Euclid’s Elements — Book XIII, Proposition 17

Constructing a Dodecahedron and Inscribing It in a Sphere

In Book XIII of Euclid’s Elements, Proposition 17 describes how to construct a regular dodecahedron, inscribe it in a given sphere, and prove that the side of the dodecahedron is the irrational straight line called an apotome.

This is the culmination of Euclid’s exploration of the Platonic solids. While earlier propositions dealt with the tetrahedron, cube, octahedron, and icosahedron, here we encounter the dodecahedron — a solid with 12 pentagonal faces that deeply connects geometry with the golden ratio.

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Proposition Statement

To construct a regular dodecahedron, inscribe it in a given sphere, and prove that the side of the dodecahedron is the irrational straight line called apotome.

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Geometric Setup

1. Begin with a cube inscribed in the given sphere.

2. Label two adjacent faces of the cube as ABCD and CBEF, meeting at right angles.

3. Bisect several cube edges to locate key points (G, H, K, L, M, N, O).

4. Use these points to determine edges on which pentagonal faces will be constructed.

5. From chosen points, cut certain line segments in the extreme and mean ratio — i.e., divide them according to the golden ratio.

6. Construct perpendiculars and equal-length segments from the cube’s surface to position new vertices outside the cube.

7. Connect these vertices carefully to form equilateral pentagons — the faces of the dodecahedron.

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Step 1 — Constructing a Pentagon on a Cube Face

• Starting from points R, S, T obtained by cutting key edges in the golden ratio, Euclid constructs a pentagon UBWCV:

  • It is shown to be equilateral.

  • It lies entirely in one plane.

  • It is equiangular (all interior angles equal).

• By repeating this construction on all 12 faces of the cube, a complete regular dodecahedron is formed.

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Step 2 — Inscribing the Dodecahedron in the Sphere

• Using relationships between the cube, its diagonals, and the pentagonal faces, Euclid proves that:

  • All vertices of the dodecahedron lie on the surface of the same sphere that circumscribes the cube.

  • Thus, the dodecahedron is perfectly inscribed.

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Step 3 — The Edge of the Dodecahedron Is an Apotome

Finally, Euclid proves that each side of the dodecahedron is an irrational straight line called an apotome:

1. Since certain cube edges are rational but divided in the extreme and mean ratio, the resulting segments are incommensurable in length.

2. From Book X, a line formed by subtracting from a rational straight line another rational straight line commensurable in square only is called an apotome.

3. Because the dodecahedron’s edges are derived from such divisions, each edge is an apotome.

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Key Results

• Construction: A regular dodecahedron is built from a cube using golden-ratio divisions.

• Inscription in a Sphere: The dodecahedron is perfectly inscribed within the given sphere.

• Edge Length: Each side of the dodecahedron is an irrational apotome.

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Modern Formula

In modern terms, if the given sphere has a circumradius $R$, the edge length $a$ of the regular dodecahedron satisfies:

$$a=R\cdot \sqrt{\frac{10-2\sqrt{5}}{3}}\mathrm{.}$$

However, Euclid expresses this geometrically:

the edge length is not rational, but belongs to the special class of irrationals called apotomes.

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Summary

Euclid constructs a regular dodecahedron, inscribes it in a given sphere, and proves that:

• The dodecahedron has 12 pentagonal faces, 20 vertices, and 30 edges.

• The vertices lie perfectly on the sphere.

• The edge of the dodecahedron is the irrational straight line called apotome.

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Why This Matters

This proposition is the culmination of Euclid’s construction of the five Platonic solids.

It shows how geometry, symmetry, and irrationality are interconnected:

• The golden ratio governs the pentagonal faces.

• The apotome emerges naturally from dividing rational lengths.

• The dodecahedron’s symmetry ties directly to the sphere.