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Aristotle was Right!
Posted:
Jun 14, 2010 3:01 PM


[ republished from mathteach / Math Forum, one typo fixed, added screen shot to Sommerville's derivation of the Mite ]
The subject heading for this thread, 'Aristotle was Right!', refers to a longstanding debate in the literature. If you Google on 'Aristotle' and 'tetrahedron', you'll find a prevalent meme: that for thousands of years people mistakenly believed that tetrahedra fill space, because Aristotle said so.
Indeed, twas the questioning of revered (churchcertified) ancient authorities that resulted in the Renaissance mindset, marked the end of what, in retrospect, many came to call a "dark age" in western civilization.
Pointing out this error of Aristotle's therefore comes across as a story with a moral: questioning authority is healthy, and not doing so may delay human progress for thousands of years.
However, if you dig more deeply into the debate, you will find that Aristotle's apologists have often cited the fact that he didn't say "regular" i.e. the "pyramid" to which he may have been referring could have been somehow irregular in shape.
This centurieslong search for spacefilling tetrahedra resulted in some pioneering studies that in themselves pushed the boundaries of what we know, right down to our own times.
Our story picks up, in the 1980s, with this article:
Which Tetrahedra Fill Space? by Marjorie Senechal Mathematics Magazine, Vol. 54, No. 5 (Nov., 1981), pp. 227243:
Majorie writes: "Aristotle did not state explicitly that he meant regular tetrahedra... some scholars continued to defend Aristotle on the grounds that he had not explicitly required regularity..."
One explorergeometer getting a lot of focus in this writeup is D. M. Y. Sommerville (18791934) who isolates what in contemporary nomenclature we call the Mite, or Minimum Tetrahedron. This is depicted in Figure 10 of the Senechal monograph, as 1/24th of the cube.
Sommerville applies two important criteria to constrain his search:
(a) the tetrahedra in question must fill space by face bonding
(b) any singular spacefiller must not rely on a mirrorimage to accomplish its spacefilling duties.
This Mite, in turn, facebonds to create two other tetra hedral spacefillers meeting Sommerville's criteria, namely the Rite (aka a tetrahedral disphenoid) and the Bite (a monorectangular symmetric tetrahedron), both classified as Sytes, i.e. those polyhedra comprised of two facebonded Mites (of which there are three, but one is a hexahedron).
So we should pause at this juncture to acknowledge that Aristotle's defenders have a strong argument: given he did not specify "regular" then his assertion is manifestly correct. Blanket, unqualified statements to the effect that tetrahedra do not fill space are manifestly incorrect.
You'll find an example of such an incorrect statement at the Math World web site, in the entry on spacefilling polyhedra:
"A spacefilling polyhedron, sometimes called a a plesiohedron (Grünbaum and Shephard 1980), is a polyhedron which can be used to generate a tessellation of space. Although even Aristotle himself proclaimed in his work On the Heavens that the tetrahedron fills space, it in fact does not."
http://mathworld.wolfram.com/SpaceFillingPolyhedron.html
(note also that no tetrahedra are depicted in the accompanying graphics, reinforcing the misimpression given by the above sentences).
The topic of spacefilling tessellations rarely arises in contemporary K16 mathematics, largely because spatial geometry as a whole has been given short shrift. Even as our technology is getting better at sharing spatial information, our K16 curriculum has been getting visually poorer, more lexical, more algebraic, less "right brained".
Some teachers call this "flying blind on instruments" and blame the Bourbaki movement. Economic factors also play a role in that textbook publishers try to get by with old figures, discourage a lot of new graphics, especially those requiring perspective.
Animations don't fit the textbook format at all, yet today's students are brought up watching television  resulting in a severe disconnect as animations are denied them in a subject which cries out for animated treatments. Those most serious about math reform at least address this disconnect, sometimes citing McLuhan.
Perhaps we have turned a corner and entered a new chapter in that regard, given recent advances in pedagogy.
According to our spanking new volumes chart, the Mite weighs in at 1/8, the Sytes at 1/4, relative to a cube of volume 3 and a regular tetrahedron of 1. These easy whole number and/or rational volumes make spatial geometry more accessible, less intimidating. The newer terminology is also more memorable. We call this a "concentric hierarchy" of polyhedra and note the long and venerable history behind it, including Kepler and many others (a NeoPlatonist tradition).
Per earlier posts to this archive (mathteach) we trace our Mite and related modules back to the five Platonics and the combinations of their duals.
Dissections of these shapes, by means of simple and logical cuts, provide the derivation for our A & B particles, along with the Mite (comprised of two As and 1B). These building blocks are suitable for elementary school use, with their derivations shown as projected screen animations and/or accomplished with clay, paper, other materials.
Older students have access to the algebraic and trigonometric expressions for characterizing these objects. The addition of vector mechanics and some computer programming provides a basis for our 21st century high school level geometry curriculum.
Chalkboard slogan:
Aristotle was Right Remember the Mite
Note: some math teachers have been discussing the possibility of labeling the Mite "Aristotle's Tetrahedron" in his honor, and in hopes of rectifying some of this tarnishing of his reputation that has been going on for some centuries. This may not catch on, but it's worth bringing up. You'll find more recent discussion on mathfuture (Google group) and in the blog post below.
Kirby Urner Oregon Curriculum Network Portland, Oregon
Related reading and viewing:
http://controlroom.blogspot.com/2010/06/aristotlewasright.html (usa.or.pdx.radmathnet)
http://www.flickr.com/photos/17157315@N00/4700244979/ (excerpt from Senechal, Which Tetrahedra Fill Space? 1981)



