Relative Sizes of Platonic Solids in Same Sphere

From ProofWiki
Jump to navigation Jump to search

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

Let $s \left({\mathscr C}\right), s \left({\mathscr D}\right), s \left({\mathscr I}\right)$ be the area of the surfaces of $\mathscr C, \mathscr D, \mathscr I$ respectively.

Let $v \left({\mathscr C}\right), v \left({\mathscr D}\right), v \left({\mathscr I}\right)$ 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:

$e \left({\mathscr C}\right)^2 : e \left({\mathscr I}\right)^2 = \left({PQ^2 + PR^2}\right) : \left({PQ^2 + QR^2}\right)$

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:

$\left({PQ^2 + PR^2}\right) : \left({PQ^2 + QR^2}\right) = \left({AB^2 + AC^2}\right) : \left({AB^2 + BC^2}\right)$

Hence:

$e \left({\mathscr C}\right) : e \left({\mathscr I}\right) = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$

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:

$s \left({\mathscr D}\right) : s \left({\mathscr I}\right) = e \left({\mathscr C}\right) : e \left({\mathscr D}\right)$

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:

$e \left({\mathscr C}\right) : e \left({\mathscr D}\right) = v \left({\mathscr D}\right) : v \left({\mathscr I}\right)$

But from $(2)$ above:

$s \left({\mathscr D}\right) : s \left({\mathscr I}\right) = e \left({\mathscr C}\right) : e \left({\mathscr D}\right)$

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

$v \left({\mathscr D}\right) : v \left({\mathscr I}\right) = s \left({\mathscr D}\right) : s \left({\mathscr I}\right)$

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

$\Box$


From $(1)$ above:

$e \left({\mathscr C}\right) : e \left({\mathscr I}\right) = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$

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

$e \left({\mathscr C}\right) : e \left({\mathscr D}\right) = v \left({\mathscr D}\right) : v \left({\mathscr I}\right)$

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

$v \left({\mathscr D}\right) : v \left({\mathscr I}\right) = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$

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.


Historical Note

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

Retrieved from "https://proofwiki.org/w/index.php?title=Relative_Sizes_of_Platonic_Solids_in_Same_Sphere&oldid=253350"