Pseudo-Random 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 S7-200 family PLCs, and the Allen Bradley MicroLogix family PLCs.
LCGs for PLCs
Vector Formulas for Curvature and Torsion in Three-space
I've previously extended the Frenet-Serret formulas to use
quaternions in four-space. However, it seems I've never documented
the same extension to vectors in three-space. Here then, are some
very useful formulas extending curvature and torsion as vectors
in three-space, as opposed to simple signed numbers as found in the
standard Frenet-Serret 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
Small C Program Creating Wireframe
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 non-working
replication, and then denouncing these claims as tricks.
Hunting the Elusive Pigtailed
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
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:
Enumerating Trinary Logic
- 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
Reverse Engineering Willis Linsay's Stepper
Steven Baxter and Terry Pratchett have provided a science-fiction
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.
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
illustrating a self replicating C program.
"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
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 Carnegie-Mellon.
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
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
Chodos, I believe, makes the correct assertion that the left
hand only helicity of neutrinos, and the right hand only helicity
of anti-neutrinos, 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, counter-clockwise
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
counter-clockwise motion approaching to a counter-clockwise 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 left-hand
neutrinos, and only right-hand anti-neutrinos. 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
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,
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
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) 229-232,
Eue Jin Jeong: arXiv:hep-ph/9704311 v4, 1997
Virtual Particles and Four-Space Trajectories
In four-space, 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, re-approach 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 three-space, suitable for use with the truss.c and flyby.c
programs, also referenced below.
Graeme Base is the author of Animalia, The Eleventh Hour - A
Curious Mystery and Worst Band in the Universe. These books,
officially for children, are especially appreciated by adults who
enjoy clever illustrations or puzzles. Unlike the Eleventh Hour,
which had a red envelope at the back which revealed the puzzle
solution, the real world does not usually publish answer keys.
While the dysfunctional world of "Worst Band in the Universe" had a
happy ending (as well as a nice CD!), the real world does not favor
Frenet-Serret Formulas In Quaternion Format
The three dimensional Frenet-Serret 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
Frenet-Serret equations extended to four-space. 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
Frenet-Serret 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, Int-J-Theor-Phys-V30-p537-545(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
Flyby - An Immersive, Interactive, 3D Wireframe Plotter
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/X-windows port in C with minimal dependencies.
(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.
Thank-you to Terry Sewards for pointing out a missing rho in an
earlier copy of this paper.
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 Truss-3D
program from the seventies, and Ameraal'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
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.
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.
Form Single Layer Solenoids
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!
Calculation for Mutual Inductance of Two Coaxial Loops in MKS
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
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 (YHMT-001)
Zhejiang Dongyang Changle Toys Co., Ltd.
DongYang Double Swallow Magnetic Stone Ltd.
No. 168 Xingsheng west road, Dongyang, Zhejiang, China Zip/Postal
Contact: Ms. Chenjie
Telephone : 86-579-6551-138 Fax : 86-579-6551-138
- This is a different company which sells magnetic clasps and
beads, using the 'magnetic hematite' tag. I don't (yet) have their
Shanmei Arts & Gifts Factory
No.2, Building 25, Hejie Village, Houzhai Street,Yiwu City,
Jinhua, Zhejiang, China
- Out of Germany, we have ChenYang Magnetics.
ChenYang-Technologies GmbH & Co. KG
Markt Schwabener Str. 8
Products CY-SE18x60 and CY-SE16x45 (Apparently 2005 time
This site clearly calls these products polished ceramic
- 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
- 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
non-magnetic. 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/min-1856.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/min-35948.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 barium-strontium ferrite
magnet: (Ba,Sr)Fe12O19 that has the magnetoplumbite structure. The
average grain size of the ceramic is 5-10 microns, and the porosity
is 10-15%. In addition, the magnetic field strength of this
material is much larger than that of any magnetite specimen. Other
items were identified as magnetoplumbite-type SrFe12O19.
- I measured the density of twelve of the BearHaversack magnets
by using an A&D Model HJ-150 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 re-measured 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.) CY-Magnetics calls out a density of
4.8 to 4.9 for their Hard Ceramic Ferrite materials.
- CY-Magnetics' 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 FH-520 (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 half-magnetized N27 neodynium
Conclusion - This really great magnetic material is not
hematite. It happens to be a really great magnetic material,
probably barium-strontium 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.
Acquire and Test Capacitors from EEStor
This is a small, Cedar Park based company which claims a process
for making barium titanite capacitors with an energy density
comparable to batteries. I want them to do well, but I what I
really want, are samples.
My business plan for this company is to set up manufacturing lines
in the former Motorola Ed Bluestein facilities, as well as the
former Dell Topfer facility in North Austin. I would use Celestica
and Flextronics (formerly Solectron) for power module assembly. I
would also hire ACC electronics students as interns, and graduates
as full employees. (This, of course, is just me talking my
The Answer is 42.
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
When we extend to traditional octonions, we have
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 |
Left Hand Octonion Unit Vector Multiplication Table
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 space-time (four true
dimensions), will require sedenions.
| 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.
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 Reed-Muller 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
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
non-predestination, 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 re-examine fundamentals of division
algebras. I am re-evaluating the works by Hamilton, Cayley,
Kirkland, Clifford, and other great mathematicians from the
1840-1890s, 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
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)
- Octonions are associated with three dimensional spaces. There
are nineteen degrees of binary freedom, giving rise to 2^19
- 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 non-distributive
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
The general antisymmetric symbol is such an excellent tool.
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
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
The Three Curvatures
Quaternions, Four Dimensional Spacetime, Frenet-Serret Equations
with Vector Curvatures
These notes show the derivation of the node coordinates for a
tetrahelix, then look at whether tetrahedrons can form
mathematically closed hoops.
Tetrahelices are a chiral structure made from tetrahedrons. I
learned about these structures playing with my children's
Extending Classical Mechanics to Allow Acceleration and Jerk in
Angular Velocity and Angular Momentum from Different Points of
Quantum Mechanics and Fourier Transforms
Tesla Coils as Transmission Lines
for Tesla Coil Transmission Lines
Transmission Lines, Reflection, and Terminations
Lab Notes on Coax
My Retro-Linux project is at http://www.StaticRamLinux.com/
Due to spam, no e-mail address is listed. However, first name dot
last name at domain is a fair bet.