The original version of the essay below was first published to geometry-research at the Swarthmore Math Forum in April of 1998.

Beyond Flatland:
Geometry for the 21st Century

by Kirby Urner
First posted: April 22, 1998
Last modified: October 29, 1998

PART I: Pascal's Tetrahedron

Agglomerating equi-radiused spheres outwardly from a central sphere in concentric cuboctahedral layers gives an isotropic Barlow packing more commonly known as the face-centered cubic or fcc. The packing subsumes the tetrahedral arrangement used to stack cannon balls, called the 'brass monkey' by some in USA civil war days, and commonly used for fruit-stacking in grocery stores. The nuclear sphere, plus layers out to one of the cuboctahedron's 8 trianglar facets, defines a tetrahedral packing.

ve5f.gif - 14.7 K

This packing is the volumetric analog of a triangulated peg board, sometimes used as in the game of pachinko to vector falling balls, with Pascal's triangle describing 'by how many paths' a given peg might be reached, with 'how many' = 'the likelihood of ending up somewhere' -- turns out to be a Gaussian distribution or bell-shaped curve.

        1 1
       1 2 1
      1 3 3 1
     1 4 6 4 1

By the same token, a tetrahedral packing, with an apexial entry point and fall-pattern with 3 choices per peg, to the base corners of a tet, defines "Pascal's Tetrahedron" of countable pathways, again with a kind of peak-shaped statistical outcome.

    1     1     1        1          1
         1 1   2 2      3 3       4  4
              1 2 1    3 6 3     6 12 6
                      1 3 3 1   4 12 12 4
                               1 4  6  4 1

Pascal's Tetrahedron also describes the structure of diamond crystals, where the carbon atoms arrange in stacked tetrahedra.

Another feature of the fcc pegboard is the fact that any four pegs not in the same plane define a tetrahedron of whole number volume, relative to a unit tetrahedron defined by four close-packed fcc spheres. This is true for skew as well as regular tetrahedra.

If the spheres have unit radius, then each sphere center is distance 2 from 12 surrounding centers, but this 2 also = 1 interval or 1 diameter, so the unit volume tetrahedron is likewise a model of 1x1x1 or 1^3, measuring in intervals.

Successive layerings of 1,3,6,10... spheres (as in Pascal's Tetrahedron) present a tetrahedron of 0,1,2,3... intervals along an edge respectively, and its volume of 1^3, 2^3, 3^3..., demonstrates the third power growth rate of volume with respect to linear distance increase. Triangles may likewise be used consistently to model 2nd powering; instead of "squaring" and "cubing" we might say "triangling" and "tetrahedroning".

t9001.gif - 4.1 K

The volume of a tetrahedron, calibrated by this unit-volume tet of 1-interval edges, may be computed using an expression similar to one obtained by Euler: only the 6 edge lengths are needed as input.


PART II: The Octet Truss

The most isotropic Barlow packing of 12 spheres around every 1 in a cuboctahedral conformation, serves as a basis for explorations in crystallography. Considered as a skeletal arrangement of edges, all length 2 and interconnecting adjacent unit radius sphere centers, we get a spaceframe known to engineers as the octet-truss. Alexander Graham Bell was among the first to study the octet-truss circa 1907 for its relevance to large scale structures, such as towers and giant kites. Fuller made use of the same spaceframe in his geometric investigations, naming it the isotropic vector matrix.

veocta.gif - 13.3 K

The octet truss defines tetrahedral and octahedral voids, twice as many of the former, with a volume ratio of 1:4 respectively. A second octet truss interpenetrates the first by centering all its vertices on octahedral voids. A third and fourth take up half the tetrahedral voids apiece.

The four trusses may be paired to give two sets of vertices arranged in a cubic pattern, ala XYZ, with one set having vertices at the centers of the other's cubes. This pattern is known as body-centered cubic or bcc in crystallography.

As Kepler once discovered, spheres packed in an fcc lattice may expand to become rhombic dodecahedra, thereby filling all the interspheric voids. An octet truss is a skeletal spaceframe with all edges perpendicular to rhombic facets. The spheres inscribed in the rhombic dodecas "kiss" at these face centers.

The rhombic dodecahedron's 14 vertices occupy the centers of the 8 tetrahedral and 6 octahedral voids surrounding any fcc sphere. Its volume is 6 relative to the tetrahedron's, thereby providing the beginnings of our concentric hierarchy:

              Shape            Volume
              -----            ------
              Tetrahedron        1
              Cube               3
              Octahedron         4
              Rh Dodecahedron    6
              Cuboctahedron     20

          Table 1: Concentric Hierarchy

The volume 3 cube inscribes as the short diagonals of the rhombic dodecahedron and runs between the pair of octet trusses in complementary sets of tetrahedral voids.

The volume 4 octahedron inscribes as the long diagonals of the rhombic dodecahedron and defines the vertices of a fourth octet truss paired with the first i.e. the one defined by the rhombic dodecahedron centers.

rhomb.gif - 16.6 K

The above concentric hierarchy shapes may all be fractured into "common denominator" modules, a minimal set of which consists of the A and the B mods, irregular tetrahedra with left and right mirrored versions (or inside-out versions), both of volume 1/24.

Left and right A mods, plus a B mod of either hand, make a minimum tetrahedral space-filler or MITE of volume 1/8. All MITEs (MInimum TEtrahedra) are outwardly identical and interchangeable, regardless of the internal B's handedness. Eight MITEs make a Coupler of volume 1, another space-filler. Our more complete hierarchy now looks like this:

modslide.gif - 9.8 K
              Shape            Volume
              -----            ------
              A module           1/24
              B module           1/24
              MITE               1/8
              Coupler            1
              Tetrahedron        1
              Cube               3
              Octahedron         4
              Rh Dodecahedron    6
              Cuboctahedron     20

          Table 2: Concentric Hierarchy


PART III: The Jitterbug

The 20-volumed cuboctahedron embeds in the fcc lattice and is defined by 12 unit-radius spheres packed around a nuclear sphere. Considered as a wireframe with flexible joints, it twist-contracts in either a clockwise or counter-clockwise direction by bringing pairs of vertices along the diagonals of its square faces closer together. When all edges are length 2, the resulting icosahedron's volume is about 18.51 relative to the cuboctahedron's of 20.

jbug.gif - 31.7 K

This transformation has been known since at least Newton's day and in the 20th century was picked up and popularized by Fuller, who dubbed it "the jitterbug" and continued its contraction to an octahedral phase and beyond. The jitterbug forms a useful bridge between 2.3.4-fold symmetric lattice shapes and the 2.3.5-fold symmetric world of non-periodic conformations.

The jitterbug transformation may be computerized using STRUCK, a Java application for building structures using edges which push or pull exponentially when forced away from their pre-defined rest-lengths. STRUCK also permits sets of edges to smoothly alter their rest-lengths through time, as in this case of the jitterbug, where the diagonals of the cuboctahedron's six square faces go from 2 x root(2) to 2 (and onward to zero at octaphase). STRUCK optionally writes successive animation frames in POV format for ray-tracing and movie-making purposes.

The fcc cuboctahedron is a subclass of fcc polyhedra (those with vertices aligned with the sphere centers in an fcc packing) known as Waterman polyhedra. Waterman polyhedra are those convex hulls consisting of all fcc vertices equidistant from a common fcc center. Because all such shapes may be tetrahedralized, we know their volume will be a whole number relative to our unit volume tetrahedron of 4 fcc spheres. Erickson has studied Watermans using STRUCK and Povray while Russell Towle, another geometer- explorer (and Povray aficionado), employed Mathematica to visualize these shapes.

shell7.gif - 5.5 K

Although the five-fold world is aperiodic, its constituent polyhedra are still amenable to modularization. Various schemes exist, among which one of the most ingenious is Koski's. Koski's mods derive from the golden cuboid, a brick with edges phi, 1 and 1/phi. These 3 edges plus distinct face and body diagonals give a set of 7 lengths, any six of which may be used in a tetrahedron. Given these base measuring cups, Koski allows each to grow or shrink by powers of phi, and finds the five-fold shapes have both algebraic and geometric equivalence to sums of such modules.

Some five-fold symmetric shapes, the 30-faceted rhombic triacontahedron for example, may be tightly shrink-wrapped around the unit-radius fcc sphere, as is Kepler's four-fold symmetric rhombic dodecahedron. An interesting fact about the shrink-wrapped rhombic triacontahedron is how closely it misses having a volume of precisely 5. By shrinking its unit radius by a hair (~0.0005) we can make each of its 120 T-modules have a volume of precisely 5/120 or 1/24, the same as the A and B modules discussed above -- a useful mnemonic for fitting this five-fold symmetric shape into our growing concentric hierarchy.

The T-mod is also the principal Koski mod, which he recursively disassembles into smaller and smaller phi-scaled versions of itself, down to his arbitrarily small "remainder tets".

rtsph.gif - 8.4 K
r1sph.gif - 9.6 K
              Shape               Volume
              -----               ------
              A module             1/24
              B module             1/24
              T module             1/24
              MITE                 1/8
              Coupler              1
              Tetrahedron          1
              Cube                 3
              Octahedron           4
              Rh Triacontahedron   5
              Rh Dodecahedron      6
              Icosahedron         18.51...
              Cuboctahedron       20

               Table 3: Concentric Hierarchy

With the above concentric hierarchy in mind, a geometry student can look at an octet truss and superimpose an easily memorable system of scaled shapes. Both four- and five-fold symmetric members are represented, with a bridging transformation, and are concentric and hierarchically arranged. Many of the shapes are also dual pairs, which pairs may then be combined to give additional shapes. For example, the cube and octahedron are duals, and combine to give the rhombic dodecahedron.

Given the streamlining effects of merging fcc packing with the octet truss and a concentric hierarchy of easy-to-remember volumes, the essentials of this curriculum are likely to gravitate down to lower grade levels, such that all of the above will be in some form accessible to an average 14 year old. Using TV, the internet, and film, it should be possible to communicate this primary level information quickly to a fairly large and global audience. We hope to have this job completed or well underway by the end of 1998.


Return to Topics for Exploration
oregon.gif - 8.3 K
Oregon Curriculum Network