1.
The two atomic triangles
Plato notes (54a1) that there is only one kind of isosceles right trianglenamely, the
45°/45°/90° trianglewhereas there are “infinitely many” kinds of scalene. But of these,
he tells us, “we posit one as the most excellent” (54a7), one “whose longer side squared
is always triple its shorter side” (54b56). Plato describes the same scalene triangle,
equivalently, as “one whose hypotenuse is twice the length of its shorter side” (54d67).
(The angles of this triangle are thus 30°/60°/90°.)
I’ll call the 30°/60°/90° triangles “
a
triangles” and the 45°/45°/90° triangles “
b
triangles.”
a
triangle (scalene, 30°/60°/90°)
b
triangle (isosceles, 45°/45°/90°)
2.
Construction of “faces” of particles out of the atomic triangles
o
Each face is either an equilateral triangle (
t
) or a square (
s
).
o
Equilateral triangles (
t
’s) are made of
a
triangles.
o
Squares (
s
’s) are made out of
b
triangles.
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 Fall '09
 JorgeRigol
 Geometry, Pythagorean Theorem, Atom, Hypotenuse, triangle

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