Eisatopon Math AI Challenges: Euclid’s Elements

Παρασκευή 26 Σεπτεμβρίου 2025

Euclid’s Elements — Book XIII, Proposition 18

Comparing the Sides of the Five Platonic Solids

In Book XIII of Euclid’s Elements, Proposition 18 concludes Euclid’s study of the five Platonic solids — the tetrahedron (pyramid), cube, octahedron, icosahedron, and dodecahedron.

In this final proposition, Euclid compares the side lengths of these solids when they are all inscribed in the same sphere.

Proposition Statement

To determine the sides of the five regular solids that can be inscribed in a given sphere, and compare them with one another.

Step 1 — Relating the Solids to the Sphere

Euclid begins with a sphere of diameter AB and uses constructions involving semicircles, perpendiculars, and ratios to establish precise relationships between the diameter of the sphere and the edge lengths of the five Platonic solids.

Step 2 — Rational Relationships

For three of the solids — the tetrahedron, octahedron, and cube — the relationships are rational:

Tetrahedron (pyramid):
$A B^{2} = \frac{3}{2} \cdot A F^{2}$
Octahedron:
$A B^{2} = 2 \cdot B E^{2}$
Cube:
$A B^{2} = 3 \cdot F B^{2}$

From these results, Euclid deduces:

The square on the side of the tetrahedron is double the square on the side of the cube.
The square on the side of the tetrahedron is 4/3 of the square on the side of the octahedron.
The square on the side of the octahedron is 3/2 of the square on the side of the cube.

Thus, the edges of these three solids are in rational ratios.

Step 3 — Irrational Relationships

The remaining two solids — the icosahedron and dodecahedron — involve irrational edge lengths:

Icosahedron:
Its edge equals the side of a pentagon inscribed in a circle whose diameter is rational.
From Proposition XIII.11, this side is the irrational straight line called minor.
Dodecahedron:
Its edge arises from cutting rational lines in the extreme and mean ratio (the golden ratio).
From Proposition XIII.17, such a segment is an irrational straight line called an apotome.

Therefore:

The sides of the icosahedron and dodecahedron are irrational.
They are not commensurable with the edges of the other three solids.
They are also not commensurable with each other: minor ≠ apotome.

Step 4 — Relative Magnitudes

Euclid proves one additional important fact:

The edge of the icosahedron is greater than the edge of the dodecahedron.

This follows from comparing proportional relationships derived from earlier propositions and the geometry of the pentagon and golden ratio.

Final Relationships Among the Five Solids

Solid	Faces	Relationship with Sphere’s Diameter $AB$	Nature of Edge
Tetrahedron	4	$AB^2 = \tfrac{3}{2} \cdot AF^2$	Rational
Cube	6	$AB^2 = 3 \cdot FB^2$	Rational
Octahedron	8	$AB^2 = 2 \cdot BE^2$	Rational
Icosahedron	20	Involves minor	Irrational
Dodecahedron	12	Involves apotome	Irrational

Mathematical Significance

This proposition summarizes and unifies all of Euclid’s work on the Platonic solids:

The tetrahedron, cube, and octahedron are rationally related.
The icosahedron and dodecahedron involve special irrationals — the minor and the apotome.
The construction highlights deep connections between:
- The golden ratio,
- Regular pentagons,
- And the symmetry of the sphere.

In modern notation, for a sphere of radius $R$ , the edge lengths $a$ are given by:

Cube: $a = \dfrac{2R}{\sqrt{3}}$
Tetrahedron: $a = \dfrac{4R}{\sqrt{6}}$
Octahedron: $a = R\sqrt{2}$
Icosahedron: $a = R \cdot \sqrt{\dfrac{10 - 2\sqrt{5}}{5}}$
Dodecahedron: $a = R \cdot \sqrt{\dfrac{10 - 2\sqrt{5}}{3}}$

Summary

Euclid compares the edge lengths of the five Platonic solids inscribed in the same sphere:

Tetrahedron, cube, and octahedron: edges in rational ratios.
Icosahedron: edges are irrational, of type minor.
Dodecahedron: edges are irrational, of type apotome.
The edge of the icosahedron is greater than that of the dodecahedron.