# A Case Study in EthnoMathematics

**Profiling subcultures by tracing memes such as “IVM”**

*Ethnography includes exploring how different ethnic groups employ mathematical concepts and skills, in everyday life and in teaching next generations.*

Bridging mathematics to art is somewhat cliche as is lamenting their coming apart and going their separate ways.[1]

The common wisdom is the “A” in STEAM (enhanced STEM) is for “Art”, not “Anthropology”.

Either way, the idea behind the STEM-to-STEAM pun is to restore that more liberal sense of “well rounded” although there’s some irony in “steam” sounding even closer to “engineer” somewhat defeating the liberalizing agenda.

We want those currently in the pipeline, perhaps toiling through “hollow schools” to have at least some vitamin “A” in their diet. [2]

If the “A” were for “Anthro”, then our over-arching goal would be to reconnect mathematics with ethnic studies. The renewed relationship with Art would arise more as a welcome side-effect of our endeavor.

We’d be getting mathematics back together with philosophy by that route too, as philo became a lot more anthropological after Russell, starting with Wittgenstein who commented on Frazer’s *Golden Bough*, and wrote of “tribes” and their “language games” in his *Remarks on the Foundations of Mathematics*.

Let me now turn to describe an ethnic artifact with moving parts, as an exhibit within this field of ethnomathematics.

Projective geometry is already full of such moving gizmos, though if they represent physical objects too closely, this might become physics instead, the domain of physics engines.

Our gadget (Koski’s and mine) is a “book” of only one page, its two covers, and the page itself, all equilateral triangles.

Open this book flat on its back, like a butterfly with its wings open, and assign all edges, including the shared spine, the length two.

The altitude of each book cover, spine center to tip, will be √3, per Pythagoras.

[ I don’t say “square root” necessarily (sometimes I do) as to insist on “square” is to put one’s thumb on a scale in a way we’re encouraging our anthropologists to detect (a kind of sensitivity training, good professional development) ]

As the page (picture it stiff) flaps back and forth, arcing from cover to cover, aligning perfectly with each, it describes two complementary tetrahedrons, with this page their common face.

One governing variable is the slant of the page from 0 to 180, and actually just climbing to vertical (90) gives all possible unique pairs, of equal volume, equal altitude tetrahedrons.

There’s plenty of trigonometry in here, and opportunities to compute. This “toy” is of pedagogical significance, within tribes that use it that way.

But what tribes work out with this gizmo to build trig skills?

That’s a question I’ll be cycling back to, as someone interested in trends, and as a curriculum developer (Google on *Oregon Curriculum Network*).

Lets posit a curriculum wherein I get to work out with just such a TetraBook type, and wherein I’m enabled to vary each of several geometric dimensions, to have the others co-vary in accordance with the above described strict constraints.

For example, if I dial in the dihedral angle, the slant of the page vs-a-vs the front cover, the two edge lengths S (for shorter) and L (for longer), from each cover tip to page tip, will both change as a consequence.

So will volume, obviously. For both tetrahedrons, it’s the same.

If you assign a possible length of S or L, the other co-varies accordingly.

S and L are themselves two legs of a right triangle with their hypotenuse our tip-to-tip length, the rhombus long diagonal. It crosses the book’s spine (its short diagonal) at its mid-point.

In keeping the page between 0 and 90, I don’t have to worry about “shorter” becoming “longer” past the 90 degree mark, so S and L work as names, for the purposes of this demo.

Lets throw in altitude as another parameter that co-varies as a dependency, but that may also be directly set.

Here’s me test-driving an instance of such a TetraBook object and checking out the default beginning configuration:

me: t = TetraBook()

me: t.long_edge

it: 3.4641016151377544

The page is flush against a cover, extending the longer edge to √12, the long diagonal of our two-book-covers rhombus.

In these dialog fragments, prompts beginning with “me:” show what I’m typing on my keyboard, whereas lines starting with “it:” show responses coming back from the computer.

For instance, here’s me yanking back on the “long edge”, retracting it from 2 √3 to √6, and thereby forcing the page to stand up vertically:

me: t.long_edge = rt2(6)

me: t.xyz_volume

it: 0.9999999999999999

Interesting, about the volume: this right tetrahedron, page vertical, has a volume of precisely 1 (our computer is sketching in limited precision base 2 numbers, but we can also use algebra).

Lets double check that the page is vertical by getting the dihedral angle on either side.

I’d like my angle measures in degrees please.

me: t.radians = False

me: t.angle_s

it: 89.99999914622636

Yep.

OK now lets set the short edge to precisely 2 so that one of the complementary tetrahedrons is regular, all edges the same length:

me: t.short_edge = 2

me: t.xyz_volume

it: 0.9428090415820638

No surprise there. The page tip has arced downward, reducing altitude. Of course the XYZ volume has dropped from 1 to something smaller.

Now lets check another attribute:

me: t.ivm_volume

it: 1.0000000000000004

What’s this?

Our TetraBook is keeping track of volume according to two different conventions, with a constant of √(8/9) between then.

Either the right tetrahedron is our unit volume (page vertical) or the regular one is (all edges 2). We switch back and forth a lot, in this curriculum. Both measures stay relevant.

The method names I was using are *xyz_volume* and *ivm_volume* respectively, naming each for a space-filling scaffolding we might both call home: XYZ and IVM respectively.

We all learn about the XYZ coordinate system in high school math classes, but may not learn about the IVM — especially that name for it — until college, or unless we happen to go to a Quaker school.

The crystallographers call it the FCC (face-centered cubic) lattice, while in the mathematics of sphere packing, it’s the scaffolding corresponding to the

CCP (cubic close packing). [3]

Alexander Graham Bell was super into this space-frame. He built towers and “kites” from it around the turn of the last century. [4]

Why the term “cube” is so prominent in both FCC and CCP takes some explaining, as the scaffolding itself consists entirely of regular tetrahedrons and octahedrons, in a ratio of 2:1 (relative frequency), with the tetrahedron

exactly one fourth the volume of the octahedron.

We find no cube-shaped cells anywhere in this IVM at the sphere diameter level, whereas XYZ is nothing but cubes of edges R. Easy to tell the difference even though they’re both isotropic and isometric in their own way.

One way to relate them is through edge lengths R and D, radius and diameter of a sphere. D-length rods build the IVM while R-length rods build the XYZ scaffolding.

This approach brings the unit volumes of each within about 6% of each other or √(9/8).

“IVM” may not be seen much outside of certain branches of American literature, associated with Hugh Kenner especially.

Some crystollographers may have given a nod in its direction, following the lead of Dr. Loeb, who taught at MIT and Harvard. Something to follow-up on and document, in ethnic studies.

The Quaker schools, such as Earlham College, are getting it from me. I’ve been using “IVM” for years for this matrix, as have Russell Chu and David Koski, lots of folks I know (Bonnie DeVarco, Amy Edmondson…).

In anthropology, we study the various nomenclatures to define our “tribes” or “ethnicities”. How do they spread? Which groups adopt the same terms? Historians use many of these same meme-tracing techniques.[5]

When it comes to dissecting the regular unit tetrahedron and volume 4 octahedron of the isotropic vector matrix, who among us talk about A & B modules? S & E modules? Anyone? We’re doing a survey, taking a poll.

A few of us raise our hands. We all have our shop talks. An A module is 1/24th of a regular tetrahedron, comes as left and right handed. A left or right B module combines with a left and right A to make the MITE, a space-filling tetrahedron (one of the space-fillers noted by Sommerville).

We can study the tests teachers use, look at websites. The outlines of a “tribe” or “ethnicity” may percolate to the foreground when we use these techniques.

Going by “outward dress” or “costume” or even “accent” is relatively less likely to distill into sensible narratives, cogent stories. Follow language. Trace memes.

What curricula feature my TetraBook? Not many so far. I just saved it to Github a couple days ago.[6]

David Koski suggested it as a curriculum artifact around 2013, well before our visit with Magnus Wenninger in 2015.[7]

Implementing co-varying dimensions of geometric figures using Python is a current thread on edu-sig at Python.org.

I’ve done a Circle and Triangle type as well, including for sharing with middle schoolers in an after school program (*Coding with Kids*).

OK, I think I’ve given some of the flavor of what the subject of ethno-mathematics looks like.

Memetics is a lot about tracing memes and in the above example we’ve seen both meme (mathematical artifact) and connected jargon (“IVM”).

You’ll find the IVM (also “octet-truss” — once patented) a recurring motif in architecture, once you know what you’re looking for.

As anthropologists, amateurs always welcome, we investigate how mathematics shows up within myriad tribal lifestyles, be that the lifestyle of a world traveling geek, or that of a high school mathematics teacher tied to a specific geographical area.

Not that these two can’t be a part of the same lifetime, as I’ve been both in my day.

As we discuss the present and future of education, let us not forget to apply an anthropological perspective.

All math is ethno-math.

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Links

[1] https://en.wikipedia.org/wiki/A_Mathematician's_Lament

[2] http://mathforum.org/kb/thread.jspa?threadID=2838432

[3] http://mathworld.wolfram.com/CubicClosePacking.html

[4] http://mashable.com/2016/02/10/alexander-graham-bell-kites/

[5] https://medium.com/@kirbyurner/american-literature-101-224489c26f19

[6] https://github.com/4dsolutions/Python5/blob/master/tetrabook.py

[7] http://worldgame.blogspot.com/2011/06/visiting-geometer-priest.html