Projects
Notes on 4x4 Magic Squares
Magic squares are usually presented as a mathematical recreation. These squares have
a sequence of integers, arranged such that the sum across each row, sum down each
column, and the sum across each diagonal equals the same magic constant. Within the
family of 880 4x4 magic squares is a subset of 48 with an even greater degree of
symmetry. These pandiagonal 4x4 magic square sum to the magic constant along every
broken diagonal, glide path and 3x3 cell spacing, yielding 52 different equations.
This note lists these sums, and shows the geometric maps associated with each sum.
Pandiagonal 4x4 Magic Squares
Direct Product Factors for Minkowski Matrices
Sixteen 4x4 matrices can be used to represent the basis for Minkowski geometric algebra.
These matrices are outer products of a set of 2x2 planar Euclidean geometric algebra matrices. This note lists the
4x4 matrices, and their originating 2x2 matrices.
Direct Product Factors for Minkowski Matrices
Symmetries In Minkowski Geometric Algebra
Inside Minkowski Geometric Algebra with signature (+,+,+,) are six subspaces with are isomorphic
to three dimensional Euclidean geometric algebra. As each of these subspaces has the same mathematical
structure as three space,
we can use our standard tools, such as translations, scaling, rotations, cylindrical coordinates,
spherical coordinates, and so forth in these subspaces.
A program demonstrating and verifying these six representations is
Six_Euclideans.ginac.cp
Furthermore, within Minkowski geometric algebra are twelve nontrivial remappings, allowing
our additional fourspace standard tools of rotations, boosts, and hyperbolic geometry to be
used in novel forms, as well.
A program demonstrating and verifying these twelve representations is
Twelve_Minkowskis.ginac.cp
Symmetries In Minkowski Geometric Algebra
Extending Geometric Algebra to Include Internal Orientation
Geometric Algebra in 3D Euclidean space can be implemented at least six
ways using 4x4 real matrices, using half the available degrees of freedom.
Digging into the multiple representations, we find 3D Euclidean geometric algebra
to be an even grade subset of a 4D Minkowski space with signature (+,+,+,).
Usually, we stop here, and get excited about 3D space as a spinor subspace of 4D.
However, I wanted to dig deeper, looking further into the symmetries
associated with this 4x4 matrix set of representations. What I came to realize,
is that we have nine elements which square to unity, which can be arranged into
a 3x3 grid such that across each row we have a set of anticommutating elements which can be used
as a linear basis, and down each column, likewise, we find sets which can be used
as a linear basis. In the ordinary geometric algebra implementation, we indeed use one of
these sets for our spatial basis and associated trivector. We use another set for our
spacetime bivectors, and the third set for three trivectors. However, the symmetries involved
give no special assignment among our six candidate arrangements. Looking at this from the
point of view of modelling an electron as a general purpose multivector, I find a neat opportunity.
Instead of viewing this 4x4 set of basis as a Minkowski spacetime, I look at this as a conventional
three space Euclidean algebra, augmented with two more displacements and associated trivectors.
These displacement may be independent deltas from a particle center, or may be internal degrees of freedom
somewhat like isospin. I prefer the real space delta, at this time. Given a three component model for the electron
of a luminal charge between two real magnetic monopoles of opposite polarity, this mathematical reinterpretation
is exciting.
Oriented 3D GA
Twelve Real Dirac Matrices
The Dirac matrices can be interpreted as a set of basis for geometric algebra
in a Minkowski metric space. The imaginary factors in Dirac's gamma matrix
have always annoyed me. In this note, I present twelve alternatives to the Dirac
matrices, using simple real numbers, for a metric (+,+,+,). Supporting code is
at Minkowski.c
Twelve Real Dirac Matrices
The Dirac Equation in Geometric Algebra
David Hestenes, Chris Doran, Anthony Lasenby, and others
have implemented the Dirac equation for the the relativistic electron in
geometric algebra. David Hestenes, in particular, has shown how the
geometric algebra view provides insight into electron motion.
This note starts with Dirac's development, then repeats the geometric algebra translation
using a (,,,+) signature. Finally, the same translation is carried out using a (+,+,+,)
signature. The result is a purely real multivector Dirac Equation with no imaginaries.
The Dirac Equation in Geometric Algebra
The Stern Gerlach Apparatus
The Stern Gerlach experiment demonstrates separation of components of a molecular beam
via the force on magnetic dipoles in a magnetic field gradient. Most presentations of
the Stern Gerlach experiment are simple cartoons or pleasant animations, which leave
the real student hungry for detail. Fortunately, several schools, such as MIT, University of Wisconsin,
Singapore National University, University of Potsdam, University of Zurich, and RWTH Aachen
have replication of the Stern Gerlach experiment part of the standard physics curriculum.
MIT and University of Wisconsin have locally produced replications, while the other schools
I looked at were using variations on a trainer made by Phywe GmbH.
(Phywe link requires cookies to be active.)
There are two major differences
between the original Gerlach Sterns versus the modern replicants.
First, the modern replications use potassium,
rather than silver, as the atomic source. Like silver, potassium has a single unpaired
electron in the outer shell. Potassium (63.4 C) has a much lower melting point than silver (962 C),
with the result that a 12W power supply is adequate for the furnace, and third degree burns are harder to obtain.
However, potassium is more chemically reactive than silver, and requires
attention in handling,
operations and cleanup. (Glauss' work also has a nice picture of the magnetic pole faces.)
The second major difference is that instead of a passive witness plate, as used in the
original experiment, PHYWE uses a LangmuirTaylor surface ionization sensor. In this detector,
neutral potassium atoms hit a heated tungsten filament, and become ionized. These ions are
accelerated by a 50V field, and impact a collector. Picoamp beam currents are routinely measured
over a small area. This device does not provide a two dimensional
display of beam intensity. Instead, a horizontal traverse is made by moving the sensor. This one
dimensional slice is only available in one plane.
University of Zurick has a nice writeup, also showing the potassium oven as well as the LangmuirTaylor probe.
A few more links.
A pleasant, slightly technical article
from the Lindau council about Stern and Gerlach. This article has photos of the original apparatus, as well
as some technical dimensions.
Another pleasant popular article, this from Physics Today, with some technical information.
Lisa Felker's Senior Project
at RWTHAachen, done concurrently with Benjamin Glauss, referenced above. Like the lab manual from Phywe, and Benjamin Glauss' senior
project, this has nice mathematical detail concerning the beam interaction with the divergent field, and nice
quality curves for beam current profiles at different magnet current settings. From the data presented, these various authors
calculate the magnetic moment for the potassium atom within a few percent of standard values.
The Lorentz Field
In the geometric algebra presentation of classical electromagnetics, we have
a multivector whose vector portion is the electric field, whose bivector portion
is the magnetic field, and whose scalar portion is a measure of deviation from
the Lorentz gauge. In this note, I propose viewing this Lorentz field Phi as of
equal status as the electric and magnetic fields. I provide wave equations for
E, B, and Phi, and point out the coalescent nature of the
Lorentz field. This presentation is not done in geometric algebra, but rather
conventional vector notation to allow easy comparison to standard texts, such
as Griffiths.
Lorentz Field in Electromagnetics
Pauli Equation and Geometric Algebra
This is the second set of notes looking at fundamental equations of quantum mechanics,
Wolfgang Pauli was adamant about keeping the spin axis distinct from the coordinate
space. Perhaps he was influenced by considerations of four dimensional space, where we can have two independent,
orthogonal planes of rotation. Perhaps he viewed the electron as a point particle, and felt that a zero dimensional
point particle, lacking extent, could not rotate, generate a magnetic field or possess angular momentum.
Unlike Louis de Broglie, Pauli was not open to the idea of an electron in continuous motion (action), selfinteracting,
following a curved path even in the absense of external magnetic fields. Regardless of how he felt, the
spin matrices he used in the development of the Pauli Schrodinger equations are a direct match for geometric algebra.
Interestingly enough, all traces of the sigma matrices disappear from the actual set of wave equations.
In effect, the sigma matrices and the geometric algebra were but a scaffolding to incorporate spin. Having served its purpose,
it is quietly removed. From this point of view, I think I can better understand Wolfgang Pauli. Slightly restating his position,
he made no assumptions about the nature of spin. It could be local geometric circulation. It could be some isospace separate from
our experience. From the point of view of the wavefunctions, the origin does not matter.
Pauli Equation and Geometric Algebra
Schrodinger's Equation in Geometric Algebra Format
I'm starting a series of notes following the steps of Hestenes, Doran and others
in writing quantum mechanics in geometric algebra form. This first note is
rather simple. The spinless Schrodinger equation is merely a complex field,
scalar plus trivector. I think the better value of this particular note is
the step by step translation to operator format for a single particle in an
electromagnetic field.
Schrodinger's Equation in Geometric Algebra Format
Extending Electromagnetism
These note explore the extension of electromagnetism by use of the geometric and wedge products
with multivector potentials.
Extending Electromagnetism via Geometric and Wedge Products
Wedge Products and Differential Forms
These are crib notes covering the use of the wedge product with derivative and differential terms. I am fascinated by
the fact that the dimensionality of space interacting with the wedge product limits the complexity we can have in our
mathematical descriptions.
Derivative and Differential Wedge Multivector Products.
Various Geometric Products
As I gain more experience with geometric algebra, I revisit my formula
summary sheets, and update them with my current understanding.
This is an update formula and code dump for three dimensional Euclidean geometry.
Code is at Tests.GA3DE_3_0_0.ginac.cp.
Geometric Product, Wedge Product, Regressive Product and Various Unary Operators.
Dimensional Analysis of Multivector and Wedge Products
Geometric algebra defines composite structures of scalars, vectors, bivectors and higher order
terms. When two multivectors are multiplied, if the internal units are different, such as meters
for the vectors, meters squared for the bivectors, and so on, then the resulting product
has mixed units in some of the product terms.
By contrast, when using the wedge product, the resulting
product has the same spatial unit arrangement as the two incoming factors.
Executive summary:

Multivector product mixes units. Plan to use dimensionless quantities, or uniform dimensioned
quantities when using the general multivector product.

Wedge product preserves dimensionality in the product.
Mixed Units in Geometrical Products
Electromagnetic Duality in 3D Geometric Algebra
This is a posting looking at duality in three dimensional electromagnetism, from the point of view of Geometric Algebra.
I present the standard Maxwell Equations, followed by a brief discussion of parity, and the axial versus polar vector
problems. I then present the standard dualities of electromagnetism, leading to the complex number
format for the Maxwell equations. Next is a presentation of geometric algebra in three dimensional Euclidean
space. I then finish with the Maxwell equations translated into multivector format, and discussion of
duality within this format.
Electromagnetic Duality in 3D Geometric Algebra
Idempotents and Nilpotents Illustrated Using Modular Numbers
This note follows Garret Sobczyk's introductory material in Chapter 1 of
New Foundations in Mathematics, ISBN 9780817683849. I provide snippets
of C code, and a step by step walkthrough of the process of obtaining the
idempotents and some nilpotents in a modular arithmetic system.
Idempotents and Nilpotents Illustrated Using Modular Numbers
A C program demonstrating these techniques is
Modular_Numbers_7.c
Exercises with Nilpotents and Idempotents
This is a set of online notes and exercises with Nilpotents and Idempotents. Some
items of interest to me are generating different nilpotents from using sandwich products
around a multivector, as in z_{new} = z*M*z, generating nilpotents using commutating
terms in geometric algebra, such as z_{new} = z*(a + b e_x e_y e_z), and identifying
factors of zero, as in P_+ P_ = 0 => P_+ = A z, P_ = z B.
Exercises with Nilpotents and Idempotents
Extreme Utility of Nilpotents and Idempotents
Idempotents (mathematical objects which square to themselves) and Nilpotents (mathematical objects which square to zero)
are powerful mathematical tools found in quantum mechanics, as well as in algebraic descriptions of spacetime.
This note discusses idempotents and nilpotents in the context of fundamentals of mathematics, and illustrates
their utility in finite dimensional spacetime, as well as their role linearizing complex operations.
Extreme Utility of Nilpotents and Idempotents
Nilpotents and Idempotents in 2D and 3D Euclidean Geometric Algebra
I present general formulas for Nilpotents (nonzero expressions which square to zero) and
Idempotents (expressions which square to themselves) in 2D and 3D Euclidean Geometries. Rather
than being simple a number/vector recombination, it proves that idempotents can easily have
scalar, vector and bivector components. Formulas are provided for general idempotents annihilator
pairs in 2D and 3D.
Nilpotents and Idempotents in 2D and 3D Euclidean Geometric Algebra
Garret Sobczyk's 2x2 Matrix Derivations
Using matrices to represent geometric algebras is known, but not necessarily the best practice. While I
have used small computer programs to scan through candidate matrices to find
representations, I have not been able to derive such matrices from first principles.
Garret Sobczyk, however, has.
I repeat his derivation here, for 2x2 matrix representation of the 2D geometric algebra,
including more intermediate steps for easier understanding.
Garret Sobczyk's 2x2 Matrix Derivations
Onward to the Fourth Dimension!
A funny thing happened on the way to the fourth dimension. . .
So I have been thinking about the relationship between Clifford Algebras and
matrix representations of these algebras. My initial belief has been that
the matrix representations are a useful, but coincidental mapping. However, looking
at the two dimensional case, I began to suspect that there may be more than coincidence
in the mapping. Specifically, the two dimensional case showed a relationship between
rotary transformations and linear translations, where each of the four algebra elements
was both an element and an operator.
As I examined the three dimensional Geometric (Clifford) Algebra, it became clear
that 3x3 matrices could not represent 3D Euclidean Clifford Algebra due to the absense of
bivectors which square to negative one. However,
in 4x4 matrices, six fundamental, and at least 48 representations were readily available. Now, in each of the
representations, each element has a nonzero determinant, which is inherently necessary for the
higher order multivector products to have nonzero determinant. This in turn is related to the
highest order multivector term having a hypervolume equal to the determinant of it's representation.
In effect, from the unit scalar, through vectors, through bivectors, including the trivector, each
representation for the three dimensional elements is algebraically a quadvector in fourspace.
From the coincidental, but useful mapping point of view, this is no big deal. From the literal
realities point of view, this is interesting indeed. Even more fun, is the categorizing of the
six fundamental representation as two independent sets of three, with the two sets dual to the other.
Regardless of set identification, the idea of three dimensional reality as a sort of downward projection of
a four dimensional greater reality is fascinating.
As I scan the matrix space for 4x4 matrices, I find 21 matrices sets (+/ factors between pairs)
with zero trace, and nonzero determinant which square to one. In a similar fashion, I find only six pairs
(+/ factors between pairs)
which square to negative one. In Euclidean Geometric algebra, vectors and quadvectors square to one, while
bivectors and trivectors square to negative one. Given the single scalar, four vectors, six bivectors,
four trivectors, and single quadvector structure of this algebra, we see that faithful representation is
impossible using 4x4 matrices, as we need ten independent elements which square to negative one, but only
six are available.
(Updated Jan 20, 2016.) We have an interesting set of 4x4 matrices, listed below, which implements Minkowski spacetime.
In the matrices below, I have 16 mutually orthogonal matrices. Ten of these, being the unit matrix, and nine others,
square to +1. The nine nonunit matrices are mutually anticommutative, and their pair wise products
produce six other orthogonal matrices, which square to 1.
Rank Order Matrix Printout
q
[ 1 0 0 0 ]
[ 0 1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]
x y z t
[ 0 1 0 0 ] [ 1 0 0 0 ] [ 0 0 0 1 ] [ 1 0 0 0 ]
[ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 1 0 0 ]
[ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ]
[ 0 0 1 0 ] [ 0 0 0 1 ] [ 1 0 0 0 ] [ 0 0 0 1 ]
y z z x x y t z t y t x
[ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 1 0 0 0 ] [ 0 0 0 1 ]
[ 1 0 0 0 ] [ 0 0 0 1 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ]
[ 0 0 0 1 ] [ 1 0 0 0 ] [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 1 0 0 ]
[ 0 0 1 0 ] [ 0 1 0 0 ] [ 0 1 0 0 ] [ 1 0 0 0 ] [ 0 0 0 1 ] [ 1 0 0 0 ]
x y z x y t z x t y z t
[ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 0 1 0 ] [ 0 1 0 0 ]
[ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 0 0 1 ] [ 1 0 0 0 ]
[ 0 1 0 0 ] [ 1 0 0 0 ] [ 1 0 0 0 ] [ 0 0 0 1 ]
[ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ]
x y t z
[ 0 1 0 0 ]
[ 1 0 0 0 ]
[ 0 0 0 1 ]
[ 0 0 1 0 ]
Source code Study_The_Nine.c
My current plan is to do some comparisions between the 2D Euclidean Algebra, the 4x4 simple representation, and the 2x2 'folded' representation.
The fact that the 2x2 matrix representations require nonzero determinants seems to put each of the representation matrices on a same footing,
as far as dimensional units are concerned.
After this, I intend to ponder the many Minkoswki implementations in 16x16 straightforward matrix representation, as well as in the 4x4
representations. I would also like to better understand why 4x4 matrices don't support Euclidean four space, but do support Minkowsi.
Finally, I want to spend some more time pondering the possibility of Minkowski space being an assembly error, where a bivector was mistakenly
used for time. At the very least, I should be able to get a nice scifi shortstory. At the best, there may be something really good here.
3D Euclidean Geometric Algebra and Matrix Representation
We cannot represent 3D Euclidean Geometric Algebras
using 3x3 matrices. The determinant of the negative unit matrix is
negative, and cannot be obtained as a square of 3x3 matrices with real
elements. Consequently, we must venture into the 4x4 matrices to represent
our three dimensional space of interest. As each 4x4 matrix has 16 degrees
of freedom, and we only require 8 for our 3D multivectors, we anticipate
multiple valid representations.
Here is a list of six representations for 3D Euclidean Geometric Algebra
using 4x4 matrices. Another 42 are easily made by sign permutations
on the x, y and z vectors.
Set 1
1 x y z
[ 1 0 0 0 ] [ 0 1 0 0 ] [1 0 0 0 ] [ 0 0 0 1 ]
[ 0 1 0 0 ] [1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ]
[ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 1 0 0 ]
[ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 1 0 0 0 ]
xy zx yz xyz
[ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ]
[ 1 0 0 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 0 0 1 ]
[ 0 0 0 1 ] [1 0 0 0 ] [ 0 1 0 0 ] [ 1 0 0 0 ]
[ 0 0 1 0 ] [ 0 1 0 0 ] [ 1 0 0 0 ] [ 0 1 0 0 ]
Set 2
1 x y z
[ 1 0 0 0 ] [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ]
[ 0 1 0 0 ] [ 0 1 0 0 ] [ 1 0 0 0 ] [ 0 0 0 1 ]
[ 0 0 1 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [1 0 0 0 ]
[ 0 0 0 1 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 1 0 0 ]
xy zx yz xyz
[ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 0 0 1 ]
[1 0 0 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 0 1 0 ]
[ 0 0 0 1 ] [1 0 0 0 ] [ 0 1 0 0 ] [ 0 1 0 0 ]
[ 0 0 1 0 ] [ 0 1 0 0 ] [1 0 0 0 ] [1 0 0 0 ]
Set 3
1 x y z
[ 1 0 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [1 0 0 0 ]
[ 0 1 0 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 1 0 0 ]
[ 0 0 1 0 ] [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ]
[ 0 0 0 1 ] [ 0 1 0 0 ] [ 1 0 0 0 ] [ 0 0 0 1 ]
xy zx yz xyz
[ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 1 0 0 ]
[ 1 0 0 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 1 0 0 0 ]
[ 0 0 0 1 ] [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 0 1 ]
[ 0 0 1 0 ] [ 0 1 0 0 ] [1 0 0 0 ] [ 0 0 1 0 ]
Set 4
1 x y z
[ 1 0 0 0 ] [ 0 1 0 0 ] [ 1 0 0 0 ] [ 0 0 1 0 ]
[ 0 1 0 0 ] [1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 0 1 ]
[ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 1 0 0 0 ]
[ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 1 0 0 ]
xy zx yz xyz
[ 0 1 0 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 0 0 1 ]
[1 0 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ]
[ 0 0 0 1 ] [ 0 1 0 0 ] [1 0 0 0 ] [ 0 1 0 0 ]
[ 0 0 1 0 ] [1 0 0 0 ] [ 0 1 0 0 ] [1 0 0 0 ]
Set 5
1 x y z
[ 1 0 0 0 ] [1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 0 1 ]
[ 0 1 0 0 ] [ 0 1 0 0 ] [ 1 0 0 0 ] [ 0 0 1 0 ]
[ 0 0 1 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 1 0 0 ]
[ 0 0 0 1 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 1 0 0 0 ]
xy zx yz xyz
[ 0 1 0 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 0 1 0 ]
[ 1 0 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 0 0 1 ]
[ 0 0 0 1 ] [ 0 1 0 0 ] [ 1 0 0 0 ] [1 0 0 0 ]
[ 0 0 1 0 ] [1 0 0 0 ] [ 0 1 0 0 ] [ 0 1 0 0 ]
Set 6
1 x y z
[ 1 0 0 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [1 0 0 0 ]
[ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 0 1 0 0 ]
[ 0 0 1 0 ] [ 0 1 0 0 ] [1 0 0 0 ] [ 0 0 1 0 ]
[ 0 0 0 1 ] [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 0 1 ]
xy zx yz xyz
[ 0 1 0 0 ] [ 0 0 0 1 ] [ 0 0 1 0 ] [ 0 1 0 0 ]
[1 0 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] [ 1 0 0 0 ]
[ 0 0 0 1 ] [ 0 1 0 0 ] [ 1 0 0 0 ] [ 0 0 0 1 ]
[ 0 0 1 0 ] [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ]
To identify vectors, which square to positive one, I wrote a program treating each cell of the 4x4
matrix as a location which could be 1, 0 or +1. In effect, this defines each possible matrix as a
sixteen digit trinary number. In this fashion, I identified 5436 matrices which square to one. To be
more selective, I then further required each candidate have a nonzero determinant,
and twelve zeroes.
This reduced the vector list to 76, but looking at the list, there were a number of entries with
trace 2 or 2, which would not be orthogonal to the unit vector. Consequently, I further restricted
the list by requiring trace = 0, which lead to the short list of 42 vector candidates. (Douglas Adams fans
should take note.) Of these 42 entries, half are negatives of the other half, so we really have 21 unique entries.
Next, I formed all 21 x 21 products, looking for antisymmetric pairs whose product squared to negative one.
These products can be either bivectors or trivectors, depending upon the context of the basis vectors chosen.
Examing the feeders for each bivector/trivector candidate, I identified six sets of positive representations
for 3D Euclidean Clifford algebras.
More details are in
3D Euclidean Geometric Algebra and Matrix Representation.
2D Euclidean Geometric Algebra and Matrix Representation
When using quaternions and complex numbers, we become familiar with
two interpretations of these items, one as a number, and the other,
as an operator. A similar situation exists with multivectors and geometric
algebra. Usually, practicioners of GA are using a vectorlike notation,
and have rulebooks describing the multiplication table of elements,
and are interested in the relationships between multivectors and their
various products at an application level.
I am currently reexamining GA with an eye toward what the basis vectors
*really are*, rather than what they do. As a tool in this investigation,
I am examining the matrix representations of the basis elements, and treating
the implied coordinate transformations as a reality, as opposed to a
bookkeeping convenience.
There is somewhat of a chicken and the egg circular reference in this situation.
Because geometric algebra is associative, and has a multiplication table, we can
find a matrix representation for the elements of this algebra. Now that I am
examining these matrix elements, I begin to suspect things are the other way around.
Each basis matrix has a clear geometrical meaning, usually involving reflections or
rotations. Reflections, in turn, look like a rotation through 180 degrees in a transverse
higher dimensional space. I suspect that the elements of GA are these fundamental
rotations potentially projected from a higher space. I suspect that the associative
nature of GA is the result of the matrix representation of these fundamental rotational
elements.
2D Euclidean Geometric Algebra and Matrix Representation
Sign Conventions, Metrics, and Geometric Algebra
We have a large number of possible sign conventions for geometric algebra (2^(2^n))
, as well as a larger number different metrics (3^(2^n)).
To help form my opinions about what I like, I write small programs to explore these different
possibilities.
My previous work in this section was seriously flawed, due to careless sign errors invalidating
the products for anything other than standard numerical order. (Wrong indices were used for
calculating the sign when rearranging the printing order of the equations.)
These invalid programs have been
removed, and replaced with a correctly written set.
These program are written
in C to create generic geometrical multivector products in 4D, with choice of /+1/0/1/ for
each metric, choice of +/ 1 for each multivector component convention, and a nice
printout of the generic multivector product, as well as the standard collection of
vector*vector, vector*bivector, vector*trivector, vector*quadvector, bivector*trivector and
so on expressions.
A demonstration program in C is Cliff_4E.c
The fuller collection of programs and outputs is
Clifford Equation Makers.tar.gz
Preferred Canonical Form for 3D Euclidean Geometric Algebra
Having gained more experience with geometric algebra, I have formed an opinion
about my preferred canonical form. I prefer the cyclic bivector format, as
opposed to the ascending numerical basis format, as it clearly shows scalar
product terms in the products of vectors and bivectors. A demonstration program
using the GiNaC library is CL3R3.ginac.cp
Preferred Canonical Form for 3D Euclidean Geometric Algebra
Curvature State Multivectors in Geometric Algebra
The FrenetSerret formulas in classical 3D analytic geometry define a curve in terms of
pathlength, curvature (deviation from a line) and torsion (deviation from a plane).
These formulas easily extend to higher dimensions, with the addition of higher order curvatures.
This note takes these concepts of deviation, and uses geometric algebra to find a surprisingly
simple multivector which encodes the instantaneous state of a curve.
Curvature State Multivectors in Geometric Algebra
Two Points of View on EM Using Geometric Algebra
Geometric Algebra is a superior replacement for vector algebra. Here is a formula
dump for two versions of EM in GA. First, a three dimensional model where the
electric field is a vector field, while the magnetic field is a bivector field.
Second, a four dimensional model where both E and M fields (as well as our
apparent coordinates) are bivectors, with one axis for E in the time direction,
while both B component are spacelike.
Two Points of View on EM Using Geometric Algebra
Component Equations for Geometric Algebra
Geometric Algebra is an approach to multidimensional physics and geometry
based upon associative, noncommutative, directed geometric products. This algebra
replaces the cross product of three dimensions, and the quaternions of four dimensions,
with a much more general set of concepts.
Usually, practitioners of GA (Geometric Algera) avoid component level work, preferring
to use symbolic multiplication in the same manner as vector arithmetic operators prefer
to avoid components, using vector symbols instead.
I find for myself, however, that I need to be able to explain concepts to a computer
to be sure that I understand these concepts myself. Accordingly, I need to be able to
write code working with GA at the component level.
Below is a multiplication table and set of component equations for the Euclidean 3D
Multivector products. Each bit in the numbers below indicates a directional basis, so
the directional unit basis are e.1, e.2 and e.4 in the table below. The scalar term is
e.0, which commutes with all other terms. The bivector terms (directed areas) are e.3, e.5 and e.6, and the
pseudoscalar (volume element) is e.7.
e.0 e.1 e.2 e.3 e.4 e.5 e.6 e.7
e.1 e.0 e.3 e.2 e.5 e.4 e.7 e.6
e.2 e.3 e.0 e.1 e.6 e.7 e.4 e.5
e.3 e.2 e.1 e.0 e.7 e.6 e.5 e.4
e.4 e.5 e.6 e.7 e.0 e.1 e.2 e.3
e.5 e.4 e.7 e.6 e.1 e.0 e.3 e.2
e.6 e.7 e.4 e.5 e.2 e.3 e.0 e.1
e.7 e.6 e.5 e.4 e.3 e.2 e.1 e.0
c.e0 = + a.e0*b.e0 + a.e1*b.e1 + a.e2*b.e2  a.e3*b.e3 + a.e4*b.e4  a.e5*b.e5  a.e6*b.e6  a.e7*b.e7 ;
c.e1 = + a.e0*b.e1 + a.e1*b.e0  a.e2*b.e3 + a.e3*b.e2  a.e4*b.e5 + a.e5*b.e4  a.e6*b.e7  a.e7*b.e6 ;
c.e2 = + a.e0*b.e2 + a.e1*b.e3 + a.e2*b.e0  a.e3*b.e1  a.e4*b.e6 + a.e5*b.e7 + a.e6*b.e4 + a.e7*b.e5 ;
c.e3 = + a.e0*b.e3 + a.e1*b.e2  a.e2*b.e1 + a.e3*b.e0 + a.e4*b.e7  a.e5*b.e6 + a.e6*b.e5 + a.e7*b.e4 ;
c.e4 = + a.e0*b.e4 + a.e1*b.e5 + a.e2*b.e6  a.e3*b.e7 + a.e4*b.e0  a.e5*b.e1  a.e6*b.e2  a.e7*b.e3 ;
c.e5 = + a.e0*b.e5 + a.e1*b.e4  a.e2*b.e7 + a.e3*b.e6  a.e4*b.e1 + a.e5*b.e0  a.e6*b.e3  a.e7*b.e2 ;
c.e6 = + a.e0*b.e6 + a.e1*b.e7 + a.e2*b.e4  a.e3*b.e5  a.e4*b.e2 + a.e5*b.e3 + a.e6*b.e0 + a.e7*b.e1 ;
c.e7 = + a.e0*b.e7 + a.e1*b.e6  a.e2*b.e5 + a.e3*b.e4 + a.e4*b.e3  a.e5*b.e2 + a.e6*b.e1 + a.e7*b.e0 ;
A simple program to build these tables and relations is Clifford_Product.c,
while a sanity check on the associative properties is Test_Euclidean.c.
A much more comprehensive list of products is at
WorkingFormulasForGeometricAlgebra.html.
As I gain more experience with geometric algebra, I am forming more opinions about style.
A short program demonstrating generic GL3 product, as well as products of vector*vector, vector*bivector,
vector*trivector, and bivector*trivector is CL3.ginac.cp.
Modelling the Electron As a 4D Oscillator
I present a toy model of the electron, consisting of an
enhanced Parson ring, with luminal circulation in orthogonal
planesets in 4D spacetime. This is a work in progress, and revisions
will occur, but I find it sufficiently interesting to post in its incomplete state.
Modelling the Electron as a 4D Oscillator
Ramanujan's Challenge
Around 1910, Ramanujan, devotee of Mahalakshmi of Namakkal, posed a number of puzzles
and challenges in the Journal of the Indian Mathematical Society , one
of which was to evaluate
\sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + 5\sqrt{ . . .}}}}}
This note walks through one solution of this puzzle, hopefully expressing my deep appreciation of
this perceptive and inspired genius..
Ramanujan's Challenge
Summertime Science Fiction and Back of the Envelope
Terry Pratchett and Stephen Baxter have released "The Long Mars", the third book in the Long Earth series.
This book is a fine bus ride companion, being little snippets easily interrupted and resumed. This book
overall is nothing exciting. The best insight from this book is that stepping spaces are gravitationally coupled,
and that the stepspace for Earth versus the stepspace for Mars do not necessarily align.
In the Universe as a Simulation theme, the Baryon Imbalance number is the ratio mismatch of the initial number
of matter versus antimatter in the initial universe. According to Ning Bao and Prashant Saraswat, this ratio is
between 6.5 to 5.9 E10. Now, having taught a few computer courses, I recognise this number as being very close to the
imbalance between positive versus negative values in 32 bit computers. We have a slight asymmetry in signed, twos
complement numbers where we can go one count more negative than positive. For 32 bit numbers, the range is
2147483647 to 2147483648, and the imbalance is 1/2^32 = 2.32830643654e10. Numerically scaling by e, to get a
predesired result, has e/2^32 = 6.32899307753e10, within the range quoted above. My scifi alterego loves the
idea that we are in a simulation, where a vendor substituted a cheaper binary processor in place of the specified
trinary processor, and we are the accidental result of this unauthorized substitution.
The concept of our universe being the interior of a rotating black hole in 5 space has gained significant traction
during the last twenty years. I am very appreciate of "The Universe is Only Spacetime" by John A. Macken for many
of the insights he presents. I especially like his treatment of the connection between gravitational potential,
and escape velocity as the paradigm to look at gravity, rather than the point of view of acceleration and
gravitation potential gradient. He points out that the escape velocity from the center of the earth is higher than
the escape velocity from surface of the earth at the north pole, which in turn is higher than the escape velocity
at the equator. Let's use this approach to the formation of a black hole in fivespace, and how it would appear to
the observers inside at later times. Being a scifi fan, I use the analogy of an illegal trash dump, where our villian
has continued to dump refuse. At the center of the dump, the escape velocity is increasing with each load until  uh oh 
we hit criticality. Treating lightspeed as a limit, the innermost radii begins to bubble outward, with the interior
empty due to the prohibition of crossing the luminal barrier. This is the big bang initiation for this new island universe.
The wavefront propagates outward at increasing speed as overlayers fall into the event horizon. This is the inflationary
epoch. Once the event horizon has consumed the mass of the illegal garbage dump, inflation stops, being starved of new
mass, and the system begins a new life as a quasi stable black hole on the outside, brand new universe on the inside.
Interior properties of the membrane are set by the mass, charge and rotation of the blackhole; this will generally be different
for each universe. As the universe really is the membrane of the black hole, dimensionality is down by one. Our five dimensional
black hole hosts a four dimensional universe. Our four dimensional universe has its own black hole decendents which
have three dimensional universes, and so on. Given that the black holes have angular momentum, one of the directional
degrees of freedom will not be symmetrical, such as our time, versus r, theta and phi. Some fun here, is that
true four dimensional space can support two orthogonal rotations spaces. . .
Cartesian Ovals
Cartesian ovals were first described by Rene Descartes, as a generalization of the ellipse.
Instead of the sum of distance from two foci being a constant, he allowed the sum to be a generalized
linear relationship. Hugyens and Newton
studied these curves for their application in diffractive optics. As we come more to the present day, we find Maxwell taking an interest in these
curves at age 14. Prior to Maxwell, and rediscovered by Maxwell, Chasle found a third focus. More recently,
and my primary source for these notes, Benjamin Harrison in the 1887 edition of his Differential Calculus, provides
a geometric study of the Cartesian ovals.
This note presents the Cartesian ovals, and identifies the focii algebraicly, and provides the explicit linear
relationships connecting the three foci.
Cartesian Ovals
From the Ellipse to the Ogg Curve
The ellipse is commonly presented in parametric form using sines and cosines. It
is easy, but wrong, to assume that this is related to polar format. This informal note
presents the parametric ellipse, then shows the conversion to CRC format polar form,
then to double angle polar form. From the double angle polar form of the ellipse, I
introduce the closely related Ogg curve, which has connections to the harmonicgeometric
mean, and elliptic integrals of the second kind.
From Ellipse to Ogg
General Mean and the ArithmeticGeometric Mean
This note first develops the General Mean for two numbers,
the investigates some variations on the AGM process.
The ArithmeticHarmonic mean converges to the simple AGM
as do any General Means with opposite signed powers. The
GeometricHarmonic means converges, but not to anything I
recognize.
The General Mean and Variations on the ArithmeticGeometric Mean
Notes on the ArithmeticGeometric Mean
These are some informal notes on the Arithmetic Geometric Mean function, and the connection to
elliptic integrals of the first and second kind. The primary references for this material
are the entertaining article by Gert Almkvist and Bruce Berndt in Americal Mathematical Monthly,
Vol. 95, No. 7 (AugSept, 1988), pp. 585608, and ``The Simple AGM Pendulum'' by Mark B. Villarino,
as well as the standard wikipedia pages for elliptic integrals.
Notes on the ArithmeticGeometric Mean
James Ivory's Historic Paper from 1796
As I delve more deeply into the history of the neglected Elliptical Functions,
I am made deeply appreciative of the Google Books scanning effort, which has
made usable copies of the scholarly journals available without travel to an
academic library. As one good turn deserves another, I have typeset James Ivory's
paper from 1796  "
A New Series for the Rectification of the Ellipsis; together with some Observations
on the Evolution of the Formula $(a^2\; +\; b^2\; \; 2\; a\; b\; \backslash cos\; \backslash phi)^n$"
This paper is significant as a predecessor to the Landen and Gauss transforms.
A New Series for the Rectification of the Ellipsis (1796)
Numerical Inverse Elliptic Integrals of the Second Kind
The elliptic integral of the second kind can be used to measure
distance along a unit ellipse, measured from the minor radius. This
function is also found in electromagnetism, as well as physics, yet
does not have a consensus defined inverse function. This is like
having arcsin, but not sine functions in your mathematical toolkit.
In the absense of a formal inverse function, several people have implemented
a simple numerical inverse function.
J. P. Boyd uses Newton's method and a clever initialization to invert elliptic integrals of the
second kind, published in``Numerical, Perturbative and Chebyshev Inversion of the Incomplete
Elliptic Integral of the Second Kind'', in Applied Mathematics and Computation (January 2012).
This approach was used by Moiseev Igor and the Google Elliptic team implementing elliptic
functions in Maple, where I first learned of this approach.
The Google function is invE = inverselliptic2(E,m,tol) at
http://elliptic.googlecode.com.
Their source code for matlab is at
http://code.google.com/p/elliptic/source/browse/trunk/inverselliptic2.m
My preference is to use c, so here is a gsl compatible, c implementation of the
inverse Elliptic Integral of the Second Kind.
GSL compatible inverse Elliptic Integral of the Second Kind in c
Length Along a Sine Curve
A simple formula for the length along a sine curve is presented, using
elliptical integrals of the second kind with imaginary parameter.
Length Along a Sine Curve
Dirac's Monopole Trick
P. A. M. Dirac's monopoles mimic radial charges, with the addition of a
singular string attached. This note presents such fields, using the lecture
notes of Professor Jose Figueroa as a starting point. I suspect that
the electric field, and Faraday's electric field lines, may have a Dirac
basis, as opposed to the radial Coulomb basis.
Dirac Radial Field from Curled Vector Potential
Electromagnetic Duality in SI Units
As noted by Heaviside, electric and magnetic fields can be transformed into
each other by an abstract rotation of 90 degrees, leaving the Maxwell equations unchanged.
Larmor extended this to a continuous rotation. However, this requires the presense of magnetic monopoles.
This note presents their work, using SI units.
Electromagnetic Duality in SI Units
Spherical Arcs Illustrated using Quaternion Division
Quaternion division is commonly illustrated using spherical arcs on a sphere.
This note turns things around, to show how to draw spherical arcs using quaternion division.
We start with standard quaternion definitions, then show the math and standard c code for drawing
great arcs on spheres. While illustrated using threespace, a very
similar extension applies to fourspace.
Spherical Arcs Using Quaternion Division
Dual Orthogonal Rotations in FourSpace
The quaternion product produces coupled rotations in orthonal planes
in fourspace, being a rotation in a (time, vector) plane as well as
a rotation in the normal threespace plane. This note presents the geometric
interpretation for unit quaternion multiplication for both pre and post
multiplication, as well as for conjugated pre and post products. The
note ends with a simple explanation for the 'sandwich' quaternion
formula for space rotations.
Dual Rotation and Quaternions
Hyperbolic Product Preserving Transforms
I want to explore scenarios where two items vary, preserving
their product. The first scenario is electromagnetism, varying
epsilon and mu. while keeping the product epsilon*mu constant.
The simplest way, of course, is to just use a factor and the
inverse. This note however, show a nice way to carry out this
operation using an angular parameter.
The Hyperbolic Transform
Exercises with Magnetic Monopoles
Magnetic monopoles are a natural extension to electrodynamics. In this set of notes,
I look at the angular momentum of the fields associated with a pure electric
charge, and a pure magnetic monopole. Using a purely classical approach, I obtain
the standard result that a pure monopole and a pure electric charge will have a component
of angular momentum independent of separation.
When integrating the angular momentum using cylindrical coordinates,
I gain an insight not seen with delta function or spherical integration approaches.
I find that half the spin of this system is concentrated in the planes between the two charges.
As the separation between charges approaches zero, this spin concentration becomes immense, being
a planar delta function.
Next, I look at duality of electric and magnetic charges under the Maxwell equation. I find that
the maximum spin for a electron/monopole mix is too small to account for the known electron spin.
This conclusion does not rule out inherent spin due to duality in the electron, but rather states
there must be other sources for spin, such as dual monopole creating a dipole, or magnetic field
due to circulation of electronic charge.
Magnetic Monopoles
The Parson Ring Model of the Electron
In 1915, Alfred Lauck Parson published "A Magneton Theory of the Structure of the Atom"
in Smithsonian Miscellaneous Collection, Pub 2371. In his paper, he proposed a spinning ring model of the electron, where the magnetic force associated with the
current ring provided the binding forces of chemical bonds. This model received attention from Arthur H. Compton, Clinton Davisson, Lars O. Grondahl, David L. Webster, and H. Stanley Allen, but fell out of favor due to the rise of quantum mechanics. This model has been periodically rediscovered, most recently by
Suichi Iida, W. Bostick, David Bergman, J. Paul Wesley, and Philip Kanarev.
This paper provides a closed form solution for the electric and magnetic potentials
of a spinning charged ring, demonstrates *why* the ring must rotate at the speed of light,
and discusses the strengths and weaknesses of the Parson model.
The Parson Ring Model of the Electron
The Voltage Potential and Electric Field of a Charged Ring
This note provides a step by step derivation of an elliptic integral closed form solution for the voltage potential and
and electric field associated with a charged filament. At the end of the note
is a listing of a quick numerical verification of closed form and discrete
sum models.
Charged Ring Potential and Field.
Homopolar Generator Design Exercises
This note illustrates the design of disk and drum style homopolar
generators using the FEMM open source finite element
solver of David Meeker. Along the way, we show the utility of the
vector A potential for modelling and explaining homopolar generator characteristics.
Homopolar Generator Design Exercises
PseudoRandom Number Generators for PLCs
Programmable Logic Controllers occasionally need random number generators
for software testing or for adaptive response. The Linear Congruential Generator
(LCG) is easy to implement, and works well. Here are implementations and discussions
of LCGs for both Seimans S7200 family PLCs, and the Allen Bradley MicroLogix family PLCs.
LCGs for PLCs
Vector Formulas for Curvature and Torsion in Threespace
I've previously extended the FrenetSerret formulas to use
quaternions in fourspace. However, it seems I've never documented
the same extension to vectors in threespace. Here then, are some
very useful formulas extending curvature and torsion as vectors
in threespace, as opposed to simple signed numbers as found in the
standard FrenetSerret formulas.
Vector Curvature and Torsion In 3D
Parametric Formulas for Villarceau Circles
At every point on a torus, four perfect circles intersect. Two of
these circles are the toroidal and poloidal circles commonly used
for coordinates on a torus. The other two circles are the
Villarceau circles, created by slicing the torus at an angle
bitangent to the interior opening of the torus. These circles can
be used as an alternative coordinate system for the torus. These
circles are also of technological interest for high frequency,
resonant, air core transformers.
Parametric Formulas for Villarceau
Circles
Small C Program Creating Wireframe
Villarceau Circles
Hunting the Elusive Pigtailed Electrocrab!
An illusionist going by the handle lifehack2012 has posted a number
of youtube videos showing small motors, lights and DVMs apparently
powered by a permanent magnet and coil combinations. Assuming a
trickster at work, we can duplicate his effects by concealed
batteries in the DVM for the DC voltage measurement, (where the
coil acts as a closing switch), concealed battery in the DC motor
(using a short rotor, or using only one of two arc magnets, (with
the coil acting as a closing switch again), and a lighting
demonstration using an open coil with hairline wires to run the
light. Naturally, an armchair approach denouncing these claims as
tricks is not as much fun as actually building a nonworking
replication, and then denouncing these claims as tricks.
Hunting the Elusive Pigtailed
Electrocrab!
Step by Step From A to B Field
Elliptic integrals are very useful for magnetic field calculations
in cylindrical symmetry. Here is a step by step example of
calculating B fields from A potentials using K(k) and E(k). Along
the way, we see some amazing cancellations in the calculations.
A to B with Elliptic Integrals
Discrete Groups via Multiplication Tables
I've always wanted to add a chapter to the textbooks using a
multiplication table approach to group theory. Nathan Carter has
provided such a tool with his Group Explorer program. In these
notes, I am using GAP and Group Explorer to illustrate the smallest
groups, with a heavy emphasis on multiplication tables, which were
the tool used by Galois, Cayley and others when developing the
theory of groups.
Discrete Groups via Multiplication Tables
Trinary Logic
What could be more fun than explicitly listing all sixteen dual
input binary logic functions? Why listing all 19683 dual input
trinary logic functions!
In this note, I list the 27 single input trinary gates, which
include three inverters, two rotators, three static levels and a
whole slew of information losers. After that, I run through 19683
different dual input trinary gates, identifying the associative,
commutative, and both associative and commutative gates. The 63
'both' gates are broken into three families of just three members,
for static level, equality and rotation gates, and then 9 families
of six members covering functions such as single trit
multiplication and modulo three addition.
Free new nomeclature:
 trinary  three level logic akin to binary.
 trit  the analog to a bit
 onefer  gates with a single output level
 twofer  gates with two of three output levels
 threefer  gates which express all three output levels
Enumerating Trinary Logic
Reverse Engineering Willis Linsay's Stepper
Steven Baxter and Terry Pratchett have provided a sciencefiction
gem with "The Long Earth". Unfortunately, the Stepper assembly
diagram by Richard Shailer has numerous errors and omissions,
making replication a frustrating experience. Likewise, an
uncredited stepper photo found on "www.io9.com" has errors and
irrelevancies. The following document attempts to improve the state
of the documentation for DIY stepping technology.
Willis Linsay's
Stepper
Interior Partitioning The Tetrahedron
We can generate embedded polyhedra by a simple technique. Between
any two vertices which make an edge, we create a new vertex for the
embedded polyhedra in the middle of that edge. Joining those new
vertices generates the new solid. For the case of the cube, this
process immediately results in a truncated cube, the cuboctahedron.
Curiously enough, we can also generate the cuboctahedron by two
stages of dividing the tetrahedron. The end of the first stage
results in an octohedron. Dividing the octahedron results in a
cuboctahedron. While the tetrahedron and octohedron are stiff
structures, the cube and cuboctahedron are not. Internally bracing
the cuboctahedron with a twelve point star does achieve a stiff
structure with cartesian orientation and stackability.
Picture and commentary
Reflections on Trusting Trust
One of the most influential essays I've read is "Reflections on
Trusting Trust", written by Ken Thompson, published in
Communications of the ACM, August 1984, Volume 27, Number 8. This
is a Turing Lecture by one of the cofounders of the C programming
language, and a significant developer of Unix. Many PDFs of this
cult classic are on the web. However, the scanned images increase
the ambiguity of the program listings provided. Here is a complete
version of his program listing
Figure 1
illustrating a self replicating C program.
To compile,
"gcc replicator.c o replicator".
To run, "./replicator > moose.c".
To verify equality. "md5sum replicator.c moose.c".
I very much appreciate Ken's admission and explanation of the
backdoor he placed into the early Unix systms, as well as his
cautions about embedded backdoors in compilers, microcode, and
hardware design.
Text To Speech for SpeakJet, VoiceBox Shield and Arduino
The SpeakJet is an artificial speech IC found on the VoiceBox
Shield for the Arduino, available at SparkFun.com. This chip really is a
phoneme generator, and needs the programmer to translate speech
into phonemes, and then into numerical codes to send to the chip
via a serial link before speech actually occurs. In a minimal
configuration, this chip requires a serial link with CTS at 9600
baud (one data wire, one CTS from CPU, as well as ground), and
provides recognizable speech. The default demo "All your base are
belong to us" was pleasant and easily understood. However, the
effort to create new phrases was rather steep. SpeakJet provides a
free application  PhraseALator  to convert text to their
numerical codes. However, words not in their dictionary are
ignored, and left to the user to provide. I really like my robots
to speak, yet I am very lazy. So, what to do?
The happy answer is to examine open source text to speech systems,
looking for existing text to speech which I can modify. One such
answer is the Festival Light (flite) software from CarnegieMellon.
After compiling their software (used in many Asterisk PBX systems,
as an aside), I use the t2p application to convert text to
phonemes. I then convert their phonemes to SpeakJet codes using a
sed script (beware  this is a Linux command line)  Flite2Speakjet.sed
 provided in the link previous. Now, the codes provided don't
modify pitch, and are not comprehensive, yet they get me much
further toward recognizable speech than the official solution
provided.
Sample sequence of commands
t2p "Now is the time for all good men to come to the aid of their
country" > moose.txt
sed f Flite2Speakjet.sed moose.txt > rawcodes.txt
I then use a text editor to place the codes into the Arduino
program, and then begin modifying by adding pitch and pauses until
I am satisfied.
Know Limits  No Limits
I was recently asked by Balu Kumar to provide a short speech prior
to the First Lego League Robotics Competition for elementary and
middle schools hosted by Westwood High School (our local high
school). I very much respect Dean Kaman, who founded First, and
National Instruments, who provided a simplified version of LabView
for the competitors. However, in general, I tend to reject any
restrictions on possible solutions for problems, such as use of
Lego kits as the only mechanical basis, rather than custom designed
components. While we have to know our limits, I prefer no limits.
This became the theme of this particular address.
Pictures are courtesy of Hubble and NASA.
Force Balanced Transmission Lines
Eric Dollard posed a transmission line puzzle.
************* Original Posting ***********************
I have a D.C. transmission line, the conductors are 2 inches in
diameter, spacing is 18 feet.
How many ounces of force are developed upon a 600 foot span of
this line, for the following;
1. 1000 ampere line current.
2. 1000 KV line potential?
Here is my answer.
Tachyonic Neutrinos and Excellent Science
I am very impressed with CERN's professionalism in verifying and
publicizing the tachyonic nature of neutrinos. I have done nothing
in this area, but I am so pleased with their work, that I want to
publicly thank these many people.
Like any major event, there have been prophets who correctly
called reality. In this case, George Sudarshan, Chodos, Hauser,
Kosteleck, Ehrlich and Eue Jin Jeong are some names to look
for.
Chodos, I believe, makes the correct assertion that the left
hand only helicity of neutrinos, and the right hand only helicity
of antineutrinos, guaranteed luminal or tachyonic speeds, and that
the presence of neutrino flavor oscillation locked out luminal only
speeds leaving strictly superluminal speeds for neutrinos. Because
his argument is so neat, I've spent some time to understand his
points, and I'm hoping to be able to communicate his arguments.
Neutrinos have an inherent spin, and consequently can be thought
of as following a spiral path as they propagate. A good mental
picture is to think of the tips of a propellor on an airplane. As
the plane flies, the tips of the propellor trace a spiral path.
Helicity is measure of the torsion of a spiraling curve. If we are
stationary, watching a plane advance toward us, counterclockwise
rotation of the propellor and the closing radial distance traces
out a right hand thread, in the sense of screw. This is positive
torsion, positive helicity. Now, assume we change our speed from
stationary to faster than the airplane. The airplane is now
separating from us, as we leave the airplane behind. From our point
of view, the trajectory of the propellor tips has changed from a
counterclockwise motion approaching to a counterclockwise motion
receding. The apparent pitch of the spiral, from our point of view,
has become negative, and is described as a left handed screw with
negative torsion (from our point of view).
We can directly measure high energy neutrinos, and we indirectly
infer high and low energy neutrinos when looking at particle
decays. The experimental fact is that we see only lefthand
neutrinos, and only righthand antineutrinos. If neutrinos
travelled at subluminal speeds, a change of reference velocity
would guarantee a mix of left and right handed neutrinos. The lower
in energy, the closer to 50/50 the randomized mix should be.
Because we see *none*, we know neutrinos had to be luminal or
beyond.
Now for the fun stuff here. Ordinarily, when we work with mass,
we are dealing with stable particles. (Think electrons, protons,
etc.) Mass for these particles is a real number, corresponding to
an inverse spatial distance in natural units. Unstable particles,
on the other hand (think muons, neutrons, etc.), get an imaginary
component of mass proportional to the inverse particle lifetime.
Particles with mass, even imaginary mass, cannot propagate at light
speed. Consequently, when the solar neutrino paradox of 1/3
neutrino flux came up, when physicist proposed neutrino flavor
oscillations, this implied neutrino mass, and that, in turned,
ruled out luminal speeds. (Neutrino flavor oscillations have been
experimentally verified using reactor generated neutrinos, emitted
as electron neutrinos, with time correlated detection of electron
and muon neutrinos at remote detectors. Japanese Kat experiment,
Minnesota experiment.)
Chodos argument from 1985 is thus: Neutrinos can't be
subluminal, can't be luminal, must be superluminal.
First verification was supernova 1987A, where neutrinos were
detected prior to optical spotting. (Found in retrospect.)
Arguments about the delayed photons propagating from the supernova
core prevented this observation from being conclusive, but
certainly provided indication. Consequently, experiments which
generated time resolved, spatially resolved neutrino beams began to
look for time of flight measurements. Fermilab MINOS measured
superluminal speeds, but the uncertainty in the measurements were
less than six standard deviations, and consequently was not deemed
definitive. CERN, in turn, has followed up, and reduced measurement
uncertainties to the six sigma standard.
Implications for future supernova detection: The supernova
events have a large neutrino pulse at fairly constant energy during
the collapsing phase transformation (flash), followed by rapidly
cooling neutrinos from the hot neutron core. As high energy
neutrinos travel slower than low energy neutrinos, (think of
proximity to light speed being the high energy condition), we will
see the time reversed rising sizzle, then flash for supernova
events. Being specific, if we see an increasing neutrino flux
coming from Betelguese or Eta Carinae, we would then later see the
high energy neutrino flash followed by the optical event.
This would be a very interesting verification, to say the
least.
Some references:
The neutrino as a tachyon, Chodos, Alan, Hauser, Avi I.,
Kostelecky, V. Alan, Phys. Lett. B150 (1985) 431.
Neutrino mass^2 inferred from the cosmic ray spectrum and tritium
beta decay, Ehrlich, Robert, Phys. Lett. B493 (2000) 229232,
arXiv:hepph/0009040.
Eue Jin Jeong: arXiv:hepph/9704311 v4, 1997
Virtual Particles and FourSpace Trajectories
In fourspace, trajectories with constant curvature, torsion and
lift are trapped on a hypersphere of fixed radius. To moving
observers, such as ourselves, as we move along the time axis, we
will see a transient disturbance as the trapped particle passes our
time plane. For particles with specific ratios of
curvature:torsion:lift, we will see a pair of particles form,
separate, reapproach and disappear. For most cases, however, we
will see a scatter of a large number of particle pairs, which last
no longer than the transit time of our time motion over the
diameter of the trapped particle hypersphere. The mathematics of
the constant curvature trajectories is given in Curves of Constant
Curvatures in Four Dimensional Spacetime. To show these curves
in four dimensions, the program RK4.c is provided. This
program compiles with any C compiler, and produces two files.
Curves.xyzt is a four dimensional coordinate set intended to be
used with hyper.c, as described below. Curves.xyt is a projection
into threespace, suitable for use with the truss.c and flyby.c
programs, also referenced below.
FrenetSerret Formulas In Quaternion Format
The three dimensional FrenetSerret formulas describe a trajectory
using pathlength (deviation from a point) as a parameter, and
specifying curvature (deviation from a line) and torsion (deviation
from a plane) as a function of pathlength. Two curves with the same
curvature and torsion histories are congruent, despite origin,
translation or rotations. It is tempting to describe physics by
having laws specify curvature and torsion. In practice, however,
the time history is essential, as different time histories and
forces can result in similar spatial curves. Consequently, if one
wants to describe physics by curvature and torsion, one must move
to four dimensional spacetime, and accept another curvature, which
I call lift, measuring deviation from a volume. It turns out that
the quaternion divisional algebra is a natural fit for the
FrenetSerret equations extended to fourspace. It also turns out
that the left handed space form has a pleasing simplicity. Even
more fun, it is easy to extend curvatures from scalar to vectors,
with an interesting alignment occuring in fourspace between
curvature and lift. Quaternion
Curvatures presents the left handed, four dimensional
FrenetSerret equations, initially scalar form, later vector
curvature form. In addition to the Frenet forms, formulas for
curvatures paramaterized by an arbitrary parameter, such as proper
time, or angular position, are provided.
Superluminal Weber Force Laws Simulations
The Weber force
law is solved for orbits using the technique shown in Clemente
and Assis, IntJTheorPhysV30p537545(1991), as well as Assis 
Weber's Electrodynamics 1994. Having verified agreement with low
speed orbits with significant angular momentum, I then go the the
extremes of tachyonic behavior for a zero angular momentum pair of
particles. The solution has repeating behaviors criticized by
Helmholtz, but which I view as fascinating. Specifically, we see
two particles collide at 1.414c, pass transparently through each
other at speed, accelerate to infinite speed in a finite distance,
change direction at infinite speed (which corresponds to zero
kinetic energy for tachyons), fall through zero radius again at
1.414c, then expand outward as classical particles, slowing to zero
speed, then repeating the fall toward zero again.This exercise does
not describe reality as we know it, as Weber's law does not use
delayed potential, nor was mass scaled relativistically. However,
this exercise has provided insight into tachyonic behavior, with a
critical speed not of c, but 1.414c.
Equivalent Magnetic Force Laws
Equivalent force laws have been a source of conflict among
electrical students since the time of Ampere, Maxwell and
Heaviside. Periodically, a new batch of enthusiasts discover
alternative force laws, but aren't aware that these force laws
cannot be distinguished using macroscopic closed current sources
for magnetic fields. To distinguish between various proposed force
law requires electron scale modelling, which is a topic for my next
posting. The pdf above provides MKS representation of five
different force laws, shows equivalent macroscopic observables, yet
different differential force elements. Source code for the open
sourced MagneticForces.c is
also provided.
Flyby  An Immersive, Interactive, 3D Wireframe Plotter
Flyby was
initially written as a Macintosh application in the 1990s, to allow
translation through a data set, as well as rotations. For stiff
problems, where multiple resonances exist on widely different
timescales, it is useful to be able to zoom in to see small
oscillations which are superimposed upon slow changes. Flyby served
me well when studying trajectories. Flyby.c is an open source,
Linux/Xwindows port in C with minimal dependencies.
Edit 23 January 2015. John Hanniball of http://anachrocomputer.blogspot.com/
noted that the previously posted version of flyby.c did not correctly
use divide by z for perspective function. I had incorrectly posted one of
my experiments using radial scaling, as with projection on a sphere rather
than a rectangular plane. I have left the radial lines commented out in the code,
and corrected the perspective function for conventional z scaling.
Thanks John!
(Compile with gcc, command gcc flyby.c o flyby lX11 lXdmcp
lXau lm )
Useful Formulas for Elliptic Integrals
Elliptic Integrals are usually tabulated in canonical form, rather
than in the form found when solving problems. Useful
Elliptic Integral Formulas Sheet is the set of formulas I find
useful when working with circular current loops.
Magnetic Field Calculation Utilities
Close form solutions for a circular current loop's magnetic fields
have been around since before Maxwell. However, the use of elliptic
integrals seems to intimidate potential users. Closed
Loop Formulas for A and B with Code is a simple example of
closed form solutions for both the vector potential "A" and the
magnetic field "B", with verifying source code for the
integrals and simulation in simple C.
Thankyou to Terry Sewards for pointing out a missing rho in an
earlier copy of this paper.
Plotting Utilities
I need simple wire frame plotters in three and four dimensions
when I'm working with trajectory simulations of fundamental
particles. The archive includes a
statically linked X11 port and source file for the *hyper* 4D
plotter and the *truss* 3D wireframe plotter, as well as sample
input files. These programs are descendents of the NASA Truss3D
program from the seventies, and Leen Ammeraal's excellent guide to
computer graphics from that same time frame.
This is a good point to advertise http://www.StaticRamLinux.com .
I very much appreciate small, simple programs and operating
systems.
Plasma Striations
Tesla coils are a popular demonstration. We often use fluorescent
bulbs and neons as interactive loads for the Tesla coils, and these
fluorescents often show a banding pattern of light and dark. Here
are a few videos demonstrating these striations, and some thoughts
about their cause.
Plasma
Striations
Single Layer Air Core Solenoids
Spring semester 2011, the students in my AC circuits class built a
variety of air core solenoids to be used as Tesla coils to light
neons and fluorescent tubes. This is a nice example of Radio Shack
hobby level technology.
PVC
Form Single Layer Solenoids
Permutation Sequences
This note, and
referenced programs Lexical.c and stdperm.c illustrate some
well known methods of enumerating all possible permutations
associated with a set.
Mutual Inductance via Elliptic Integrals in MKS Units
Coaxial circular windings can have mutual inductance and field
calculations simplified by use of complete elliptic integrals or
AGM functions. Here is a step by step classical derivation,
followed by a C numerical double integration. Both work well, but
the classical form calculates much faster. This note has been
corrected on March 6, 2011 to correct a missing 4 pi divisor as
pointed out by Clifford Curry. Thanks!
Classical
Calculation for Mutual Inductance of Two Coaxial Loops in MKS
Units
"Magnetic Hematite"
A novelty sold at local fairs are 'Snake Eggs', made in China,
billed as magnetized hematite. The first observation is that the
residual field in these magnets is quite strong, seeming comparable
to neodynium/boron magnets. The second observation is that the
material a tough, impact tolerant, ceramic. The third, is that this
material is a good insulator. This is clearly an interesting
material for permanent magnets motors. So, it is time to
investigate.
My tasks here are to
 Identify the products, sources and factories for this product.
 This is the original product of interest.
Magnetic Singing Rattle Snake Egg Magnets (YHMT001)
Zhejiang Dongyang Changle Toys Co., Ltd.
DongYang Double Swallow Magnetic Stone Ltd.
No. 168 Xingsheng west road, Dongyang, Zhejiang, China Zip/Postal
: 322118
Contact: Ms. Chenjie
Telephone : 865796551138 Fax : 865796551138
 This is a different company which sells magnetic clasps and
beads, using the 'magnetic hematite' tag. I don't (yet) have their
product.
Shanmei Arts & Gifts Factory
No.2, Building 25, Hejie Village, Houzhai Street,Yiwu City,
Jinhua, Zhejiang, China
http://www.joyjw.com http://www.joyjw.cn
 Out of Germany, we have ChenYang Magnetics.
ChenYangTechnologies GmbH & Co. KG
Markt Schwabener Str. 8
85464 Finsing
Germany
http://www.cymagnetics.com
Products CYSE18x60 and CYSE16x45 (Apparently 2005 time
frame)
This site clearly calls these products polished ceramic
magnets.
 BearHaversack.com sells `magnetic hematite' by the pound. I
bought a five pound bag of various geometrical shapes (not the
Snake Eggs shape). These appear to be the same base material, but
have parallel planes for easy magnetization.
 Ascertain the actual material used. Hematite is not naturally
magnetic.
 From http://www.webmineral.com/data/Hematite.shtml
Hematite  Fe2 O3 (Fe 3+)
Molecular weight 159.69 gm
No residual magnetic field
Reddish brown streak
So, first, easy tests are the magnetic and streak tests.
Compare with hematite ore samples from BearHaversack. The ore is
nonmagnetic. The ore streak test against white unglazed ceramic
(bottom of coffee cup) is reddish brown for mineral hematite, but
blackish grey for "magnetic hematite". These are clearly different
materials. Checking the internet for other investigations, we find
. . .
 From http://www.mindat.org/min1856.html
NOTE  the 'hematite' used in jewelery, and often sold as
magnetized items, is nothing of the sort and is an artificially
created material, see Magnetic Hematite.
 From http://www.mindat.org/min35948.html
An artificially created magnetic material (this contains NO
natural Hematite) widely sold as 'magnetic hematite' or simply
'hematite'. Please note that the name 'hematite' is quite
misleading, as this is NOT a natural stone.
Investigation of one item offered for sale as 'magnetic hematite'
showed it was composed of a ceramic bariumstrontium ferrite
magnet: (Ba,Sr)Fe12O19 that has the magnetoplumbite structure. The
average grain size of the ceramic is 510 microns, and the porosity
is 1015%. In addition, the magnetic field strength of this
material is much larger than that of any magnetite specimen. Other
items were identified as magnetoplumbitetype SrFe12O19.
 I measured the density of twelve of the BearHaversack magnets
by using an A&D Model HJ150 scale to measure weight, and a
small beaker to measure volume. To isolate magnetic effects, I used
an eight inch tall piece of styrofoam packing to keep the magnets
away from the ferrous material in the base of the scale. To verify
accurate measurement of the mass of the magnets, rather than mass
and magnetic attraction to ferrous materials in the base and
supports, the magnets were doubled over to form a quadrapole,
rather than dipole, and the magnets were remeasured in multiple
directions and orientations while observing the same measured mass.
The total mass was 180.5g, the volume (measured by water
displacement in a beaker), was 38 mL. The density is 4.75 g/mL. (As
opposed to 5.6 for hematite.) CYMagnetics calls out a density of
4.8 to 4.9 for their Hard Ceramic Ferrite materials.
 CYMagnetics' ceramic magnets have a Curie temperature of about
450 C. Red hot is about 520 C. Heating in an oven did not get hot
enough to demagnetize the sample. Heating with a propane torch
resulted in brittle failure due to too high a rate of heating.
Instrumented kiln heating looking for magnetic drop is probably
required to measure an accurate Curie temperature.
 The surface residual magnetic field was measured using a
Sypris/FWBell FH520 (177101) Hall probe. This probe has a nominal
sensitivity of 100mV/T when run at an excitation of 25 mA. I did
not do a Helmholtz calibration on this particular probe. The
particular probe has a 3.5 mV offset, and was run at 22.9 mA, for
an estimated sensitivity of 91.3 mV/T. Measurements were made with
the probe white side up and then in the same place with the probe
white side down, the numbers subtracted and divided by two to
reduce the offset voltage. The BearHaversack magnets had a residual
field of 0.17T to 0.26T, highest near edges, lower in middle of
faces. By contrast, my neodynium magnets measured 0.52T. These
numbers are consistent with a Y10 ceramic material for the
BearHaversack material, and a halfmagnetized N27 neodynium
material.
Conclusion  This really great magnetic material is not
hematite. It happens to be a really great magnetic material,
probably bariumstrontium ferrite ceramic.
Future work  I want to make samples of magnetite via the
Massurt method both for ferrofluid fun and for characterization of
magnetite's magnetic properties.
Arduino Based Gauss Level Three Axis Magnetic Sensor
I've posted a small program to read the HoneyWell HMC5843 triple
axis sensor using the I2C interface of the Arduino 2009 board. This
sweet sensor is used as a solid state compass in cell phones, and
is not really suited for high field (motors and magnets) work. It
is, however, ideal for low fields.
hmc5843.pde
Project 42
The Answer is 42.
Really.
Complex numbers, quaternions and octonions are division algebras,
where multiplication and magnitudes can be defined in such a way
that the magnitude of a product is also the product of the
magnitude of the two multiplying terms. This type of product then
allows us to define division. My fascination has been the fact that
there are so many different formulas in these two, four, and eight
dimensional spaces that also satisfy the norm relationship above.
Initially, I used brute force to enumerate working algebras,
finding 2^3 solutions for the complex family (comps), 2^8 solutions
for the quaternion family (quads), and 2^19 families for the
octonion familty (octs). These individual solutions are merely
choices for the polarity (sign) of the product terms in the
defining basis multiplication table.
The encoding of the basis vectors as binary numbers, and the
product base being given by XOR is worth illustrating numerically,
as some interesting interpretations can be made about
dimensionality and multiplication. For traditional quaternions, we
have numbers and three spatial dimensions. Borrowing notation from
spacetime, I'll call t=00 as the numbers (scalars), i=01 as one
space axis, j=10 as another space axis, and k=11 as a third. The
multiplication table is
Right Hand Quaternion Unit Vector Multiplication Table
Prefactor Postfactor  Binary format
 
 1 i j k  t*i = 00^01 = 01 = i
 \  t*j = 00^10 = 10 = j
1  1 i j k  t*k = 00^11 = 11 = k
 
i  i 1 k j  i*j = 01^10 = 11 = k
  i*k = 01^10 = 10 = j
j  j k 1 i 
  j*k = 10^11 = 01 = i
k  k j i 1 
When we extend to traditional octonions, we have
Left Hand Octonion Unit Vector Multiplication Table
Prefactor Postfactor

 1 i j k E I J K
 \
1  1 i j k E I J K t = 000 Scalar

i  i 1 k j I E +K J i = 001 Vector

j  j k 1 i J K E I j = 010 Vector

k  k j i 1 K J I E k = 011 Area

E  E I J K 1 i j k E = 100 Scalar

I  I E K J i 1 k j I = 101 Area

J  J K E I j k 1 i J = 110 Area

K  K J I E k j i 1 K = 111 Volume
Product base formed by XOR of two factors. Example i*K => 001^111 = 110 = J
Polarity (sign) determined separately.
The interesting interpretation of the above, seen in Clifford
algebra and geometric algebra, is that the quaternion table above
is *not* a four dimensional structure, but rather a two dimensional
structure, where the multiplication terms involving i and j give
rise to an areal term k. In a similar fashion, complex numbers are
really dealing with a one dimensional space, and octonions with a
three dimensional space. To get a real spacetime (four true
dimensions), will require sedenions.
Knowing that the basis multiplication table can be encoded as
binary numbers XORed together, and seeing the power of two number
of solutions, I decided that I should examine the solutions as a
digital logic problem. Given that the basis logic was XOR based, I
was pleased to find ReedMuller XOR implementations of the sign or
polarity logic found above.
Having found digital logic solutions for normed algebras in two,
four and eight dimensions, my next target was 16 dimensions.
Sedenions are known to not be normal. While I can find numerical
special cases where two integer sixteen vectors and a sixteen
vector product satisfy the norm relations (based upon any integer
being the sum of four squares), there is no general formula. Doing the sixteen
bit digital exercise, despite knowing unlikely success, led to
an interesting result. In my
approach, I used one bit to determine an active high/active low
default state for a bit. I then used higher order bits to determine
participation of free variables in the sign of the term in the
multiplication table. The interesting result, is that while I had
conflicting definitions for active high/active low default bit
states, I had a consistent set of definitions for how the default
state would be modified by 42 free
variables.
This result has me very excited. I've always wanted to find a
simple explanation for quantum superposition. My hope is to find a
simple, mathematical analog to the ring oscillator or logic
paradox, where a feedback path with an odd number of inversions,
coupled with a propagation delay through the logic creates an
inherently oscillating system. A multiplication table which is
inherently oscillatory, gives rise to an oscillatory metric, which
in turn justifies much of our experience with quantum multivalued
weirdness. Philosophically, an inherently oscillating metric
structure of space is a good model for Planck scale quantum foam,
and in the bigger picture, justification for 'free will', or
nonpredestination, on the quantum scale.
So, what do I know? I know that the base definition is
inconsistent, and that there are flaws (inherent conflicts) in my
model for the multiplication table logic. However, I also know,
that once a suitable basis is defined, I can give 2^42 new
variations on a successful basis.
My current task is to reexamine fundamentals of division
algebras. I am reevaluating the works by Hamilton, Cayley,
Kirkland, Clifford, and other great mathematicians from the
18401890s, as well as Sylvester (1867), Hadamard (1893) and Walsh
(1923) in more recent times. My most recent influence is the
geometric algebra interpretation of Clifford algebras by David
Hestenes.
My current working assumptions are
 Complex numbers are associated with one dimensional spaces.
There are three degrees of binary freedom associated with complex
numbers, giving 8 (2^3) different comps.
 Quaternions are associated with two dimensional spaces. There
are eight degrees of binary freedom, giving rise to 256 (2^8)
different quads.
 Octonions are associated with three dimensional spaces. There
are nineteen degrees of binary freedom, giving rise to 2^19
different octs.
 Each of the previous spaces have no ambiguity in the
calculation of their structure constants.
 Sedenions are associated with four dimensional spaces. Seds and higher
spaces fail to form traditional division algebras. The failure to
form static division algebras at sedenions and beyond is well known
(Frobenius and Kirkland). One work around is the nondistributive
approach of Albrecht
Pfister , kindly demonstrated in code by Warren
Smith. Another approach, that of J. R. Young (1848), is to note
that a product of 16 squares can be written as a sum of 32 squares
(using octonions), which can be mapped into complex components. My
new goal, is to use the complex definition of Young, and identify
2^42
variations.
The general antisymmetric symbol is such an excellent tool.
The
Antisymmetric Symbol
The Fibonacci numbers are deeply related to the Golden Ratio. Here
is a simple proof of the closed form equation for the n'th
Fibonnaci numbers. This formula can be extended to negative
integers, as well as treated as a continous function of n. This
continuous function has many similarities to the force law of Roger
Boscovitch.
Fibonacci Numbers
Every trajectory in space can be described by the curvature and
torsion as a function of pathlength. Pathlength differentially
measures deviation or distance from a point. Curvature
differentially measures deviation from a line. Torsion
differentially measures deviation from a plane. The circle is a
curve of constant curvature, while the spiral has constant
curvature and torsion. Extending to four dimensions, we now have
another curvature, "lift", which differentially measures lift out
of a volume. The curve of constant curvature, torsion and lift is a
trajectory on the surface of a hypersphere, consisting of
circulations at two different linear frequencies in orthogonal
plane sets.
The Three Curvatures
in Fourspace
Quaternions, Four Dimensional Spacetime, FrenetSerret Equations
with Vector Curvatures
Quaternion
Toolbox
These notes show the derivation of the node coordinates for a
tetrahelix, then look at whether tetrahedrons can form
mathematically closed hoops.
Tetrahedral Coordinate
Calculations
Tetrahelices are a chiral structure made from tetrahedrons. I
learned about these structures playing with my children's
GeoMags.
Tetrahelices
Extending Classical Mechanics to Allow Acceleration and Jerk in
Dynamical Potentials
Euler's Equations
Extended
Angular Velocity and Angular Momentum from Different Points of
View
Angular Momentum
Quantum Mechanics and Fourier Transforms
Quantum Mechanics
Tesla Coils as Transmission Lines
Telegrapher's Equations
for Tesla Coil Transmission Lines
Transmission Lines, Reflection, and Terminations
Lab Notes on Coax Cable Reflections (html)
Lab Notes on Coax Cable Reflections (pdf)
My RetroLinux project is at http://www.StaticRamLinux.com/
.
Due to spam, no email address is listed. However, first name dot
last name at domain is a fair bet.