5 platonic solids

Platonic Solids in a park in Steinfurt, Germany 

We would like to talk about 5 platonic solids.

The definition of a platonic solid is a shape where you’ve got sort of different sides to the shape, but all of the sides are the same shape. That shape is a regular kind of polygon - so like the square where all the sides are the same length and the angles are all the same and every corner looks the same.

This is called a tetrahedron. It’s faces are regular polygons, regular triangles. It has the same number of triangles meeting at each corner. Tetrahedron is like a pyramid, but unlike the pyramid - it has a triangular base, not a square base. It contains of 4 faces, 4 vertices and 6 edges.

To jest czworościan foremny (tetraedr). Składa się z 4 trójkątów równobocznych i ma taką samą liczbę ścian przystających do każdego z wierzchołków. Ma 4 wierzchołki i 6 krawędzi.

This is the octahedron. It has eight (the preffix ’oct’ means ‘eight’) triangular faces. It’s like two pyramids glued together (by the bases). Unlike the tetrahedron, this has four triangles meeting at each corner. It has got 8 faces, 6 vertices and 12 edges.

Ta bryła, to ośmiościan foremny (oktaedr). Wygląda jak dwie sklejone piramidy. Na każdy wierzchołek w tej bryle zbiegają się 4 trójkąty równoboczne. Ośmiościan foremny ma 8 ścian, 6 wierzchołków i 12 krawędzi.

This solid is called an icosahedron. You can see every single corner has got five triangles meeting at it and all of the faces are regular triangles. It has got20 faces, 12 vertices and 30 edges.

To jest dwudziestościan foremny (ikosaedr). Na każdy wierzchołek tej bryły zbiega się 5 równoboczno trójkątnych ścian. W sumie ikosaedr posiada 20 ścian, 12 wierzchołków i 30 krawędzi.

This one over here is a hexahedron (cube). A hexahedron has six square faces. There are 3 squares meeting at each corner. It’s got 6 faces, 6 vertices and 12 edges.

Sześcian ma 6 kwadratowych ścian i na każdy wierzchołek przypadają 3 z nich. Poza tym posiada 6 wierzchołków i 12 krawędzi.

Finally, this is the dodecahedron. The preffix ‘dodec’ means ‘twelve’. This solid is made of 12 pentagon faces and at each vertex we have 3 pentagons. It’s got 12 vertices and 30 edges.

Jako ostatni przedstawiamy dwunastościan foremny (dodekaedr). Składa się on z 12 pięciokątnych ścian i na każdy wierzchołek zbiegają się trzy pięciokąty. W sumie ma 12 wierzchołków i 30 krawędzi.

The first prove that Platonic Solids are one of a kind is this table.

This table contains of the sums of faces, vertices and edges of each 5 platonic solids. The magic begins when we sum up faces with vertices of a solid and from that sum we subtract the number of edges of that solid…

Ta tabelka składa się z ilości ścian, wierzchołków i krawędzi każdej z 5 brył platońskich. Spróbujmy zsumować liczbę ścian i wierzchołków każdej z tych brył a potem odjąć liczbę krawędzi… 

What’s happening? In the end it always equals 2!

Wynik zawsze wychodzi taki sam - 2.

Let’s change the topic a bit.

There’s a way to prove that there are only 5 platonic solids, and we are going to show it just now!

Let’s search for the possible shapes with traingular faces. It has to have at least 3 triangles around a point. 

There you have it. Let’s imagine this corner (the black dot). And around this corner there are 3 regular triangles. There’s a huge gap there, a space. Because of that you’re able to glue these sides together - it’s going to be a concave shape. It will be 3 triangles around the point - that makes your tetrahedron~

Let’s add another triangle! 

Now we’ve got 4 triangles around the point. There’s still gap there, so you can easily glue those sides together. If we will do that, we will end up with four triangles around a point - that makes your octahedron~

Let’s draw another triangle in here. 

So now we’ve got 5 triangles around the point. Glue it together (again, you’ve got a little missing gap) and this will make will make your icosahedron. Five triangles around the point. 

If you try to keep going, if you try six triangles around the point though, all you get is…

…a flat object. It’s just a flat piece of paper. It’s not concave. It’s not a shape. So that’s far as you can go. 7, 8 - you can’t add anymore. You can’t add any more triangles.

All right, so let’s do the squares. Let’s have 3 squares around the point. 

And these are all 90 degrees. There’s a missing gap here. Glue it together, the possible sides together - and you’ll get your cube. 

If you try to add a fourth square to it, you’ll get 360 degrees. You’ll get a flat shape again. There’s no point in that.

Now, let’s try the pentagons. 

The only thing that we have to do is to draw a single pentagon. If I draw pentagons, a pentagon has an angle of 108 degrees. You can fit 3 of those around a point. If you fit 3 around the point that would add up to 324 degrees. And then you have a little gap and you can glue it together. If you will do it you’ll get 3 pentagons around the point - that’s the dodecahedron. 

You can’t fit 4 pentagons around the point, because the angles will be waytoo big.

Let’s move on and try the hexagons. Hexagons have an angle of 120 degrees.

If you try 3 hexagons around the point, that will add up to 360 degrees. It’s going to be flat. It makes a flat shape once again. It’s not a concave shape, not a solid. After that, if you’re looking for 7-sided shapes or 8-sided shapes - the angles are going to be too big and they won’t make regular solids at all.

Podsumowując po polsku (czerpiąc przy tym z tego tekstu) możemy powiedzieć, że bryła platońska musi mieć:

• wszystkie ściany złożone z identycznych foremnych figur,
• w każdym wierzchołku zbiega się taka sama liczba krawędzi,
• wszystkie kąty wewnętrzne są wypukłe (mniejsze niż 180*).

sources: wikipedia, google 

//Julia Ziemba & Karolina Kierepka