Five Platonic Solids/Proof 2
Jump to navigation
Jump to search
Theorem
There exist exactly five platonic solids:
- $\paren 1: \quad$ the regular tetrahedron
- $\paren 2: \quad$ the cube
- $\paren 3: \quad$ the regular octahedron
- $\paren 4: \quad$ the regular dodecahedron
- $\paren 5: \quad$ the regular icosahedron.
In the words of Euclid:
- I say next that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another.
(The Elements: Book $\text{XIII}$: Proposition $18$ : Endnote)
Proof
Consider a convex regular polyhedron $P$.
Let $m$ be the number of sides of each of the regular polygons that form the faces of $P$.
Let $n$ be the number of those polygons which meet at each vertex of $P$.
From Internal Angles of Regular Polygon, the internal angles of each face of $P$ measure $180^\circ - \dfrac {360^\circ} m$.
The sum of the internal angles must be less than $360^\circ$.
So:
\(\ds n \paren {180^\circ - \dfrac {360^\circ} m}\) | \(<\) | \(\ds 360^\circ\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds n \paren {1 - \dfrac 2 m}\) | \(<\) | \(\ds 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds n \paren {m - 2}\) | \(<\) | \(\ds 2m\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds n \paren {m - 2} - 2 \paren {m - 2}\) | \(<\) | \(\ds 2m - 2 \paren {m - 2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {m - 2} \paren {n - 2}\) | \(<\) | \(\ds 4\) |
But $m$ and $n$ are both greater than $2$.
So:
- if $m = 3$, $n$ can only be $3$, $4$ or $5$
- if $m = 4$, $n$ can only be $3$
- if $m = 5$, $n$ can only be $3$
and $m$ cannot be greater than $3$.
There are $5$ possibilities in all.
Therefore all platonic solids have been accounted for.
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.)