In three dimensional space, the tetrahedron is the simplest Platonic solid. Building out tetrahedrons from faces of a tetrahedron in a systematic fashion gives rise to four different families of shapes. These assemblies are useful for sub-atomic physics, crystallography, mechanical engineering and home entertainment. The iterated construction technique is inspired by crystal growth and Wolfram's cellular automata.

The rendered images below are inspired by the very enjoyable 'geomag' toys, which unfortunately have been displaced in the toys stores by cheaper, non-functional knock-offs. Actual image rendering was done using POV-Ray. Source files samples are linked at the bottom of page.

Iterative Process

The construction process for these images begins with an initial tetrahedron defined by four points. I've imaginatively labelled these points as 0, 1, 2, and 3. We choose three of these four points to define a triangle base, and create a new tetrahedron by calculating the new apex by formula. This algorithm creates 24 different shapes, being four choices for the first point, three for the second, and two for the third. Each of the shapes can be identified by a three digit number. In the code fragment below, I generate 33 sections of the 231 structure.

Fuller Source Code

p0 = vector(0.0, 0.0, 0.0);

p1 = vector (0.5, -0.5*sqrt(3.0),0);

p2 = vector (1.0, 0.0, 0.0);

// p3 = (1.0/3.0)*(p0 + p1 + p2) + (2.0*sqrt(2.0)/3.0)*cross((p1 - p0), (p2 - p0));

WritePOV_Header();

DrawBase(p0, p1, p2); // Draw the initial base;

p3 = AddOn(p0, p1, p2);

for (i=0;i<33;i++) {

spinecolor++;

// now we try iterative construction

n0 = p2;

n1 = p3;

n2 = p1;

p0 = n0;

p1 = n1;

p2 = n2;

p3 = AddOn(p0, p1, p2);

}

Simplest Iterative Structure - Simple Tetrahedron

Iterative families 012, 023, 031, 103, 120, 132, 201, 213, 230, 302, 310 and 321 lead to the same initial tetrahedron. (This is like a unity transformation.)

The initial base is yellow, and the plaid back strut is a composite of red, blue and green elements overlaying each other with the pattern seen being due to floating point truncations.

Next Simplest Case - Back to Back Tetrahedrons

Iterative families 021, 102, and 210 lead to oscillating tetrahedrons. The first transformation creates the backing tetrahdron, and the second transformation creates the original. (This is like an inversion transformation.)

The initial base is yellow, and the plaid struts are a composite of red, blue and green elements overlaying each other with the pattern seen being due to floating point truncations.

Chiral Structures - Left Hand Helix

Looking at a tetrahedron doesn't immediately impress one with it's inherent 3D asymmetry. However, if we label faces (with a base down), we can label 123 clockwise or 123 counterclockwise. Chirality is inherent in the tetrahedron. Structures 123, 130 and 320 clearly show this left hand thread behavior. Novel features are a three strand DNA appearance, with the outer outer strands cross-braced by two different spatial frequency internal brace structures. (Note the cyan/magenta versus white frequencies.) Also note that while the red, green and blue outer strands are left hand helices, the cyan and magenta bracing form right hand helices. From a mechanical engineering point of view, this truss is quite stiff. From a science fiction writer's point of view, this could make some really interesting alien genetic storage. In terms of function, this is a left helical propagator.

Chiral Structures, Right Hand Helix

Structures 203,231 and 301 form a right helix, the 4D mirror image of above. In terms of function, this is a right helical propagator.

Circulating Disk Truss

Structures 013, 023 and 312 complete our set of possibilites with a very interesting disk structure. The angle between tetrahedral planes is 70.529 degrees (arccos(1/3)). This is not an integral submultiple of 360 degrees, so we have an almost periodic structure on repetition.

The images below show stages in the evolution of this iterated structure.

Three wedges

Four Wedges

Five Wedges.

This type of almost periodic structure features an angular mismatch that is very interesting. As we continue propagation, we get more of a disk structure.

Six wedges (blue red green sequence)

And then some more...

Summary and Teaser

These structures just looked at are beautiful on their own. These may also have an interesting tie-in with the internal structure of fundamental particles.Try to connect the chiral spirals propagators with massless particles. Try to connect the disk propagators to mass-possessing particles. Try to connect geometric lattices with particles that are bound to each other by potentials that increase with separation while simultaneously repelled by potentials that vary with inverse distance. Figure out how to cram in one more particle, and see how time figures into all of this.

Enjoy!

December 20, 2006

Sample Main program

KNComplex.h

Quaternions.h