Relative Sizes of Platonic Solids in Same Sphere

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Summary

In the words of Hypsicles of Alexandria:

If $AB$ be any straight line divided at $C$ in extreme and mean ratio, $AC$ being the greater segment, and if we have a cube, a dodecahedron and an icosahedron inscribed in one and the same sphere, then:
$(1) \quad \left({\text{side of cube} }\right) : \left({\text{side of icosahedron} }\right) = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$;
$(2) \quad \left({\text{surface of dodecahedron} }\right) : \left({\text{surface of icosahedron} }\right)$
$ = \left({\text{side of cube} }\right) : \left({\text{side of icosahedron} }\right)$;
$(3) \quad \left({\text{content of dodecahedron} }\right) : \left({\text{content of icosahedron} }\right)$
$ = \left({\text{surface of dodecahedron} }\right) : \left({\text{surface of icosahedron} }\right)$;
and $(4) \quad \left({\text{content of dodecahedron} }\right) : \left({\text{content of icosahedron} }\right)$
$ = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$

(The Elements: Book $\text{XIV}$: Proposition $8$ : Summary)


Proof

Let:

$\mathscr D$ be a regular dodecahedron
$\mathscr I$ be a regular icosahedron
$\mathscr C$ be a cube

which are inscribed in a given sphere.


Let $\map e {\mathscr C}, \map e {\mathscr D}, \map e {\mathscr I}$ be the edge (that is, the side) of $\mathscr C, \mathscr D, \mathscr I$ respectively.

Let $\map s {\mathscr C}, \map s {\mathscr D}, \map s {\mathscr I}$ be the area of the surfaces of $\mathscr C, \mathscr D, \mathscr I$ respectively.

Let $\map v {\mathscr C}, \map v {\mathscr D}, \map v {\mathscr I}$ be the volumes of $\mathscr C, \mathscr D, \mathscr I$ respectively.

Let $AB$ be cut at $C$ in extreme and mean ratio such that $AC$ is the greater segment.

From Proposition $7$ of Book $\text{XIV} $: Ratio of Lengths of Sides of Cube and Regular Icosahedron in Same Sphere:

$\map e {\mathscr C}^2 : \map e {\mathscr I}^2 = \paren {PQ^2 + PR^2} : \paren {PQ^2 + QR^2}$

where:

$PQ$ is the radius of the circle which circumscribes the faces of both $\mathscr D$ and $\mathscr I$
$R$ is the point at which $PQ$ has been cut in extreme and mean ratio such that $PR$ is the greater segment.

Thus from Lemma to Proposition $8$ of Book $\text{XIV} $: Ratio of Volumes of Regular Dodecahedron and Regular Icosahedron in Same Sphere:

$AB : AC = PQ : PR$

Thus:

$\paren {PQ^2 + PR^2} : \paren {PQ^2 + QR^2} = \paren {AB^2 + AC^2} : \paren {AB^2 + BC^2}$

Hence:

$\map e {\mathscr C} : \map e {\mathscr I} = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$

and so $(1)$ is shown to be true.

$\Box$


From Proposition $6$ of Book $\text{XIV} $: Ratio of Sizes of Surfaces of Cube and Regular Icosahedron in Same Sphere:

$\map s {\mathscr D} : \map s {\mathscr I} = \map e {\mathscr C} : \map e {\mathscr D}$

and so $(2)$ is shown to be true.

$\Box$


From Proposition $8$ of Book $\text{XIV} $: Ratio of Volumes of Regular Dodecahedron and Regular Icosahedron in Same Sphere:

$\map e {\mathscr C} : \map e {\mathscr D} = \map v {\mathscr D} : \map v {\mathscr I}$

But from $(2)$ above:

$\map s {\mathscr D} : \map s {\mathscr I} = \map e {\mathscr C} : \map e {\mathscr D}$

Thus from Proposition $11$ of Book $\text{V} $: Equality of Ratios is Transitive:

$\map v {\mathscr D} : \map v {\mathscr I} = \map s {\mathscr D} : \map s {\mathscr I}$

and so $(3)$ is shown to be true.

$\Box$


From $(1)$ above:

$\map e {\mathscr C} : \map e {\mathscr I} = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$

From Proposition $8$ of Book $\text{XIV} $: Ratio of Volumes of Regular Dodecahedron and Regular Icosahedron in Same Sphere:

$\map e {\mathscr C} : \map e {\mathscr D} = \map v {\mathscr D} : \map v {\mathscr I}$

Thus from Proposition $11$ of Book $\text{V} $: Equality of Ratios is Transitive:

$\map v {\mathscr D} : \map v {\mathscr I} = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$

and so $(4)$ is shown to be true.

$\blacksquare$


Historical Note

This proof is Proposition $8$ of Book $\text{XIV}$ of Euclid's The Elements.

Result $(3)$ appears to have been included in Comparison of the Dodecahedron with the Icosahedron by Apollonius of Perga. This work is now lost.


Sources

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