The same face, the same corner
A Platonic solid is a polyhedron in which every face is the same regular polygon, and the same number of faces meet at every vertex. A cube is one: every face is a square, and 3 squares meet at every corner.
At a corner of a solid, at least three faces must meet. Two flat faces can only meet along a single edge, where they are hinged like the pages of a book; they cannot close in around a point. Three is the fewest that can.
At every corner of a cube, 3 square faces meet: at the corner nearest you, the top, the front and the side.
Less than a full turn
Put six equilateral triangles together around one point. Each has angles of 60°, and 6 × 60° = 360°, a full turn, so the six triangles fit exactly and make a flat hexagon. A flat patch cannot be a corner of a solid.
Now take one triangle out and pull the gap closed. The remaining five triangles have to tilt up to meet, and the point in the middle rises into a corner. Closing a gap is what lifts a corner, so the angles meeting at a corner of a solid must add up to less than 360°.
Six equilateral triangles meeting at one point fill the full turn around it, 6 × 60° = 360°, and lie flat as a hexagon.
Which faces fit under 360°
Each interior angle of a regular polygon with n sides is (n − 2) × 180° ÷ n. That gives 60° for an equilateral triangle, 90° for a square, 108° for a regular pentagon and 120° for a regular hexagon.
Equilateral triangles: 3 of them make 180°, 4 make 240° and 5 make 300°, all under 360°. Six make exactly 360°, which is flat. So a corner can have 3, 4 or 5 triangles.
Squares: 3 make 270°, but 4 make 360°. Regular pentagons: 3 make 324°, but 4 make 432°. So squares and pentagons can meet only 3 at a corner.
Regular hexagons: 3 already make 360°, so no corner can be made of them. A polygon with more sides has larger angles still, so 3 of them make more than 360°, and none of those can make a corner either.
The angle sum at a corner for each face and each number of faces. Only five sums are under 360°: 180°, 240° and 300° for triangles, 270° for squares and 324° for pentagons.
Five corners, five solids
That leaves exactly five possible corners: 3, 4 or 5 triangles, 3 squares or 3 pentagons. Each of them builds a solid, and those five solids are the Platonic solids. No sixth one can exist, because there is no sixth corner to build it from.
Euler’s formula, V − E + F = 2, tells how many faces each one has. Take the corner made of 3 pentagons, and say its solid has F faces. The pentagons have 5F edges between them, and each edge of the solid is shared by 2 faces, so . The pentagons have 5F corners between them, and 3 of those meet at each vertex of the solid, so . Putting these into Euler’s formula gives . Multiply every term by 6: 10F − 15F + 6F = 12, so F = 12.
The solid has 12 faces, 5 × 12 ÷ 2 = 30 edges and 5 × 12 ÷ 3 = 20 vertices, and 20 − 30 + 12 = 2. The same working for the other four corners gives 4, 8 and 20 faces for the three corners of triangles, and 6 for the squares.
The five corners that fit under 360°, and the number of faces of the solid each one builds.
Their names
The names are Greek: hedra means a face, and tetra, octa, dodeca and icosa mean 4, 8, 12 and 20. So the tetrahedron has 4 triangular faces, the octahedron 8 and the icosahedron 20, and the dodecahedron has 12 pentagonal faces. The cube, with 6 square faces, is also called a hexahedron.
They are named after the Greek philosopher Plato, who wrote about them around 360 BC. The proof that there are only five appears at the end of Euclid’s Elements.
The five Platonic solids: the tetrahedron, octahedron and icosahedron from triangles, the cube from squares and the dodecahedron from pentagons.
Two slips
Counting faces at a corner as if they were solids. Three squares meet at each corner of a cube, but squares build only one Platonic solid, the cube. Triangles are the face of three different solids, because they can meet 3, 4 or 5 at a corner.
Allowing an angle sum of exactly 360°. Four squares around a point make 4 × 90° = 360°, and they lie flat like tiles on a floor. The sum at a corner must be less than 360°, so squares meet only 3 at a corner.
Worked example: A Set of Role-Playing Dice: the Twenty-Sided Die, and Which New Dice Could Be Made
Question A set of role-playing dice has dice with 4, 6, 8, 12 and 20 faces. On each die every face is the same regular polygon, and the same number of faces meet at every corner. (a) The 20-sided die has equilateral triangles for faces, with five meeting at each corner. Find its number of edges and its number of corners. (b) A designer wants a new die of the same kind, either with six equilateral triangles at every corner or with three regular hexagons at every corner. Explain why neither can be made, and list every regular polygon, with the number of faces at each corner, that could make a die of this kind.
1.Count the edges face by face: 20 triangles have 20 × 3 = 60 edges between them. Each edge of the die is shared by two faces, so the die has E = 60 ÷ 2 = 30 edges.
Each edge of the die is shared by two triangles: 60 ÷ 2 = 30 edges. 2.(a) The triangles also have 20 × 3 = 60 corners between them, and five of them meet at each corner of the die, so V = 60 ÷ 5 = 12. The die has 30 edges and 12 corners. Check with Euler's formula: V − E + F = 12 − 30 + 20 = 2.
(a) Five triangles meet at each corner: 60 ÷ 5 = 12 corners, and 12 − 30 + 20 = 2. 3.At a corner of a solid, the angles of the faces meeting there must add to less than 360°. If they add to exactly 360°, the faces lie flat and cannot fold up into a corner. Six equilateral triangles give 6 × 60° = 360°, and three regular hexagons give 3 × 120° = 360°, so neither can make a die.
Six triangles or three hexagons make 360°: they lie flat and cannot fold into a corner. 4.At least three faces meet at every corner. Equilateral triangles, with angles of 60°, can meet three, four or five at a time, making 180°, 240° or 300°. Squares, with 90°, can meet only three at a time, making 270°. Regular pentagons, with 108°, can meet only three at a time, making 324°.
Triangles fit three, four or five to a corner; squares and pentagons only three. 5.(b) A regular polygon with six or more sides has angles of at least 120°, and three of them make at least 360°. So the only possible dice are triangles with 3, 4 or 5 at a corner, squares with 3, and pentagons with 3: 5 shapes in all, which are the 4-, 8-, 20-, 6- and 12-sided dice of the set.
(b) Polygons of six or more sides make at least 360° with three at a corner, so there are 5 shapes of die.
Answer: (a) 30 edges and 12 corners; (b) six triangles make 6 × 60° = 360° and three hexagons make 3 × 120° = 360°, so each lies flat; the only possibilities are triangles with 3, 4 or 5 at a corner, squares with 3 and pentagons with 3, which is 5 shapes
Common mistakes
- Dividing the 60 triangle corners by 3, the number of corners of one triangle, to get 20 corners. What matters is how many faces meet at each corner of the die, which is 5.
- Thinking that hexagons are a good choice because they fit together with no gaps, as in a honeycomb. Fitting with no gap is exactly what makes them lie flat; a corner of a solid needs the angles to add to less than 360° so that the faces can fold up.
More volume and surface area problems, worked step by step →