Aristotle was Right!
Kirby Urner
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)
Kirby Urner
here's some follow-up reading:
http://mathforum.org/kb/thread.jspa?threadID=2093280&tstart=0
Warning: some Euclidean dogmas may be questioned.
Kirby
Kirby Urner
space-fillers in his 1923 paper, Space-Filling Tetrahedra.
The Mite, and the two tetrahedral Sytes it assembles, were
three of four of them.
However the Rite also fractures into tetrahedral space-
fillers that are duplications of one another, and not made
of Mites.
Bucky Fuller never wrote about it or gave it a name,
probably did not know of Sommerville's results, which
matched his own minus this additional space-filler.
In sum:
Tetrahedral space-fillers:
Mite
Bite = Mite + Mite
Rite = Mite + Mite
1/4 Rite
Sommerville thinks these are the only four to fill
Euclidean space (by face bonding and without handedness).
The volume numbers we use in using Fuller's geometry
are discussed in linked resources.
Basically: Mite 1/8; Bite 1/4; Rite 1/4; 1/4 Rite 1/16.
Follow-up summer reading (if you're not too bored by
spatial geometry):
http://worldgame.blogspot.com/2009/02/mite-fight.html
Kirby