Euclid’s Elements — Book XIII, Proposition 16

Constructing an Icosahedron and Inscribing It in a Sphere

In Book XIII of Euclid’s Elements, Proposition 16 deals with one of the most beautiful of the Platonic solids — the regular icosahedron.

Euclid shows how to construct an icosahedron, perfectly inscribe it in a sphere, and prove that the side of the icosahedron is an irrational straight line called minor.

This proposition unites the geometry of circles, pentagons, and polyhedra into a single elegant construction.

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Proposition Statement

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

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Geometric Setup

1. Begin with a sphere of diameter AB.

2. On AB, choose a point C so that:

   $$AC=4\cdot CB\mathrm{.}$$

3. Construct a semicircle ADB on AB and draw CD perpendicular to AB.

4. Set out a circle EFGHK with radius equal to DB.

5. Inside this circle, inscribe a regular pentagon EFGHK.

6. Bisect the arcs between pentagon vertices to form another pentagon LMNOP and draw auxiliary segments that will later define the triangular faces.

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Step 1 — Building the Framework

• From vertices E, F, G, H, K, erect perpendiculars EQ, FR, GS, HT, KU to the plane of the pentagon, each equal to the circle’s radius.

• Join the top points to form another pentagon QRSTU, symmetric to EFGHK.

• These two pentagons — one at the base, one elevated — form the skeleton of the icosahedron.

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Step 2 — Constructing Equilateral Triangles

Using Euclid’s earlier results:

• Triangles formed from hexagon, decagon, and pentagon side relationships ensure that all relevant faces are equilateral triangles.

• By combining these triangles carefully, Euclid creates a solid bounded by 20 equilateral triangular faces:

  • 12 vertices

  • 30 edges

  • 20 faces

Thus, the icosahedron is fully constructed.

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Step 3 — Inscribing the Icosahedron in the Sphere

• Using properties of semicircles and right angles, Euclid shows that all vertices of the icosahedron lie exactly on the surface of the sphere.

• Hence, the icosahedron is perfectly inscribed.

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Step 4 — The Side of the Icosahedron Is “Minor”

Euclid proves that the edge of the icosahedron equals the side of the inscribed pentagon.
From Proposition XIII.11, we know:

If an equilateral pentagon is inscribed in a circle whose diameter is rational, the side of the pentagon is the irrational straight line called minor.

• Since the pentagon EFGHK lies in a circle with a rational diameter, its side is the irrational line minor.

• Because the icosahedron’s edges coincide with the sides of this pentagon, each edge is also minor.

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Key Results

• Icosahedron Construction: Built from 20 equilateral triangles.

• Inscription in a Sphere: All vertices lie perfectly on the sphere’s surface.

• Edge Length: Each side of the icosahedron is the irrational straight line called minor.

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Modern Formula

If the sphere has circumradius $R$, the edge length $a$ of the icosahedron is:

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

This matches Euclid’s conclusion, but Euclid expresses it geometrically:
the edge length is not rational but of a special irrational type — the minor.

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Summary

Euclid constructs a regular icosahedron inscribed in a sphere and proves that:

• The solid has 20 equilateral triangular faces.

• Its vertices lie perfectly on the sphere.

• The side of the icosahedron is the irrational straight line called minor.

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Why This Matters

The icosahedron is one of the most symmetric shapes in nature.

Euclid’s construction shows how pentagonal geometry and the golden ratio combine to form this exquisite solid, linking it deeply with the geometry of the sphere and the special irrational numbers studied in Book X.