Martian Math

PATH versus STEM

Pretty much everyone in curriculum development in the anglophone world (English-speaking) knows about STEM: Science, Technology, Engineering, Mathematics.

Not all publishers or standards bodies accept the acronym for internal use, but will be familiar with it vicariously.

PATH, on the other hand, is more obscure. I first came across it in The Math Myth by Andrew Hacker. I use it to mean: Philosophy, Anthropology, Theater and History.

What I like about both PATH and STEM is they posit a somewhat science fiction reality, contrary to fact perhaps, yet fun to think about. My “Global University” (Global U), also somewhat fantastic (a creature of fantasy), might use these two pillars for consensus building plans.

That’s what planning is a lot about by the way: building consensus. Even if we never build Old Man River City (OMR), the stadium shaped mega-project, it served as a conversation piece and focus of attention.

Science fiction does that too, in projecting both utopian and dystopian possible futures. In this sense, science fiction is akin to investment banking. Architecture is all about creating persuasive mock-ups, helping to persuade potential stakeholders to jump in.

Mosque in Ohio

In the Media Age, heralded by Marshall McLuhan and others, we’ve come to see that Politics is Theater. Behind the public figures, giving speeches, are the people who write those speeches. Others provide briefings.

The various public figures star in a series of partially overlapping “soap operas” (dramatic productions) designed to guide the spectator public regarding how they might participate as well, as extras, as would-be stars.

Politics is also Anthropology however, as is Statistics & Demographics.

Again, PATH and STEM provide useful pigeonholes, and given only eight of them, they force a lot of conceptual integration, and that’s useful in itself.

Philosophy also partakes of Anthropology (cite Wittgenstein) and so on. We’re fine with blurring distinctions, cross-fertilizing and so on. Philosophy likes to partake of Mathematics. Pigeonholes are not impermeable containers.

Speaking of Mathematics, in the age of computer aided design (CAD), even for free in the cloud (check for example), I think it makes sense to start with spatial geometry, which is more experiential, and then work down to the plane. I involved in an after school program that shares CAD concepts with elementary and middle school aged children, so I’m prone to brainstorm in this vein. But then the reason I’m so involved has to do with pre-existing interests.

When you think of a plane, remember your viewpoint, your camera angle, is off the plane, looking at the plane. Planes are in space. Space is logically prior to planes. Plane geometry is embedded in spatial geometry, conceptually speaking. Our everyday experience is spatial, not planar.

History enters the picture when we research current curricula and how they came about. Various script and screenwriters got into the act, resulting in Theater. Competing empires played a role.

The anglophone world put a lot of stock in Euclidean geometry and that’s heritage we want to keep of course. Euclid was into spatial geometry as well, not just the planar stuff. Some of the German stuff, like “clock arithmetic” got shoved aside to make more room, because the anglophones of the day were eager to downplay Germanic heritage. Americans were being prepared to fight some wars, with Germans the “bad guys”.

We need not go into all of that here, just lets keep the Math Wars in view, and remember that the mathematics curriculum has been politicized too. The history of ideas is no simple, steady progression. STEM and PATH go hand in hand. I’m one of the screenwriters myself, one of the curriculum developers with a point of view and corresponding agenda. Whereas I put my cards on the table, others may choose to keep their hands hidden.

TetraBook: one page flaps back and forth

Imagine a database of a great many polyhedrons, categorized (tagged) according to various properties they share. Remember the Platonics? That’s a very exclusive set, of only five shapes: Tetrahedron, Cube, Octahedron, Icosahedron, Pentagonal Dodecahedron. They come as duals to one another, and duals combine to beget yet more shapes.

You may have seen my Youtubes on all this, but what I’m talking about is common knowledge. I’ll let you know when my program diverges.

Euler proved that polyhedrons, defined a certain way, always follow the rule that the number of corners plus faces equals the number of edges plus two (V + F = E + 2). Some of the proofs of Euler’s Theorem for Polyhedrons are quite elegant, however in Martian Math (as I call it) we do a lot of vocabulary building with descriptive elements before we do a lot of proving. What’s a corner? What’s a face? What’s a polyhedron? What’s a dual?

A link from planar to spatial geometry is tiling and space-filling. What tiles cover a plane with no gaps? What shapes fill space in a gapless manner? Figurate and polyhedral numbers enter in, as both tiling and space-filling relate to number sequences. The Book of Numbers by Conway and Guy explores this territory, as does Gnomon, by Midhat Gazale.

Nested Hierarchy of Polyhedrons

The polyhedrons in our database, retrieved by tag, such as Platonic, Archimedean, space-filling, convex, stellates, five-fold symmetric and so on, are likewise “objects” in the object-oriented sense. We may start programming, learning to code, with geometric concepts front and center.

Here is where CAD tools enter in as well. Selection tools allow us to choose Vs, Fs, or Es (vertices, faces or edges), in addition to whole objects. Then we might twirl our objects (spin them, rotate them), around various axes, creating corresponding networks of circles.

If you think animations would be helpful here, you’d be right. Rotation, translation, and scaling become a part of our vocabulary.

I mentioned Martian Math above, and that’s clearly divergent vocabulary. You might suppose any mathematics branding itself that way would contain alien content. You would be correct about that. If you check out some of my other stories on Medium, you’ll find out more about this curriculum, but rest assured I’ll share more details here.

color poster

The dual of the cube is the octahedron. The dual of the pentagonal dodecahedron is the icosahedron. The tetrahedron is dual to itself. We might consider the operation of “taking the dual” as a “unary operation” as we only need one polyhedron as input. That’s a lot like “taking the inverse” and I’m tempted to play with the “~” symbol for this dual-making purpose:

~cube = octahedron.
~octahedron = cube.

We could start introducing a symbolic language in this manner. The comes combining duals:

cube + ~cube = rhombic dodecahedron (RD)
~RD = cuboctahedron
tetrahedron + ~tetrahedron = cube
icosahedron + ~icosahedron = rhombic triacontahedron (RT)

You see where I’m going with this right? I’m introducing the idea of “operators”, both unary and binary. I’m encouraging a form of “mental geometry” to aid and abet what we already call “mental arithmetic”. Given our polyhedrons are programmed objects, we may actually implement these operators in code elsewhere in the curriculum.

Now I’m going to break with tradition and introduce American Transcendentalism, a school of thought. AT suggest this alternative volumes table, which is unfamiliar to most English speakers:

Volumes Table
Tetrahedron: 1
Cuboctahedron: 2.5
Cube: 3
Octahedron: 4
RT: 5
RT: 5+
RD: 6
RT: 7.5
PD: ~15.35 or (3√2)(ΦΦ+1)
Icosahedron: ~18.51 or (5√2)(ΦΦ)
Cuboctahedron: 20
RT: ~21.21 or 20 √(9/8)

This content is definitely alien. If you went to an English school, you will probably be scratching your head around now. Yes, Φ (phi) stands for the “golden ratio” or (1 + √5)/2, and ΦΦ stands for phi times itself, but why does the RT show up so many times? Long story. Mind the gap. :-D

Martian Math Artifacts

Schools influenced by American Transcendentalism are more likely up to speed on the volumes table above. The core fact to realize is we’re accounting in “tetravolumes” meaning the unit is tetrahedron shaped. The one at the top calibrates all the shapes further down.

Any shape may be scaled to any size, but there’s a logic, an order, to using this chart in particular, relating to edge lengths. They all fit around a common center and so we use the term “concentric hierarchy”. What you see above is sometimes referred to as “the Sesame Street of Synergetics” where Synergetics is a core text in the AT syllabus, by R. Buckminster Fuller (RBF) in collaboration with E.J. Applewhite (EJA). You may have seen my other stories on Medium, giving more details.

Henry Ford Museum

Clearly, in using expressions involving √ and Φ, we’re moving beyond the elementary school level, into middle and high school.

Speaking of political theater, you may be aware of some of the tensions and confusions surrounding the Turkish sufi intellectual, Fethullah Gulen. Some of the charter schools established in North America were products of the Hizmet movement he established, and which some called the Turkish Peace Corps. Like the Kennedy-established Peace Corps, Hizmet was mired in controversy as any foreign service tends to be.

Hizmet would fly politicians from Washington, DC to Turkey and show them around, a form of lobbying. On a small planet, such as Spaceship Earth, that sort of thing happens all the time, in the name of fostering greater mutual understanding.

I mention Hizmet because of the evident reciprocity between American Transcendentalism and other cultures. Earlham College in Indiana has ties to Turkey. Many universities and colleges have long standing student exchange programs.

Earlham Exhibit

The Quakers also have a service organization named American Friends Service Committee (AFSC). I’ve done some work for the latter, and support Earlham College (I’m mailing a contribution today in fact). My postings to Quaker think tanks mention Hizmet here and there.

Like I said before, I put my cards on the table and acknowledge up front that Martian Math has a political component. EJA was high up in the CIA for example, though had retired well before we became friends. His years of service in Capitalism’s Invisible Army didn’t prevent him from appreciating Dr. Fuller, his boyhood hero.

Imagine a Quaker-influenced school, with or without a charter, sharing the above Volumes Table as a matter of routine. That’s just part of what we teach, both locally and via distance education circuits.

American Transcendentalism is alive and well in this scenario.

What political elements might consider this development to be a threat, if any? I pose that question as an exercise: test your ability to anticipate future soap operas.

Quakers are pacifists and have not been particularly averse to Islam in their history. Are Quaker schools exporting Martian Math to Turkey? To Iran? To England? Quakers got started in England after all.

EJA at Crown Point, Oregon

To what extent is the anglophone world likely to be “tainted” by some newfangled alternative mathematics curriculum featuring object oriented programming, databases, and CAD? I invite researchers to study this open ended question. I don’t claim to have all the answers myself.