Projects



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 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 non-working 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:
Enumerating Trinary Logic


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.

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 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 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 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 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) 229-232, arXiv:hep-ph/0009040.

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.

Interesting Puzzle


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 independent idealists.

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 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/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.

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 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 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

  1. Identify the products, sources and factories for this product.
  2. Ascertain the actual material used. Hematite is not naturally magnetic.

  3. 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.

  4. 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.

  5. 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 material.

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.

hmc5843.pde


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 book.)


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 space-time (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 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 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 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 Hestenes.

My current working assumptions are


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, Frenet-Serret 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




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.