Date: Jun 14, 2010 3:01 PM Author: Kirby Urner Subject: Aristotle was Right! [ republished from math-teach / 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 (church-certified) 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 centuries-long search for space-filling 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. 227-243:

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 explorer-geometer getting a lot of focus in this

write-up is D. M. Y. Sommerville (1879-1934) 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 space-filler must not rely on a mirror-image

to accomplish its space-filling duties.

This Mite, in turn, face-bonds to create two other tetra-

hedral space-fillers meeting Sommerville's criteria, namely

the Rite (aka a tetrahedral disphenoid) and the Bite (a

mono-rectangular symmetric tetrahedron), both classified

as Sytes, i.e. those polyhedra comprised of two face-bonded

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 space-filling polyhedra:

"A space-filling 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/Space-FillingPolyhedron.html

(note also that no tetrahedra are depicted in the accompanying

graphics, reinforcing the mis-impression given by the above

sentences).

The topic of space-filling tessellations rarely arises in

contemporary K-16 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 K-16 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 (math-teach) 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/aristotle-was-right.html (usa.or.pdx.radmathnet)

http://www.flickr.com/photos/17157315@N00/4700244979/

(excerpt from Senechal, Which Tetrahedra Fill Space? 1981)