The 5 Elements

Part 21: Proposition

The plane cannot be filled by a congruence of the individual angles of plane figures of four or more kinds.

For the 4 smallest angles are those of the trigon, the tetragon, the pentagon, and the hexagon.

The first and last of these add up to two right angles, while the second of them is a right angle and the third is greater than a right angle, by one fifth of a right angle.

Therefore, if they are joined up they come to more than four right angles. So by XVI they do not form a congruence. We should exceed four right angles by even more if larger angles were taken.

Part 22: Axiom 22

If 2 plane angles do not add up to more than a third one, they will not form a solid angle with it.

Part 23: Proposition

2 plane angles of a figure with an odd number of sides will not come together with an angle of another kind to form a regular solid.

For by Part 17, the solid angles would not all be the same, which is not in accordance with definitions V to X.

Part 24: Proposition

3 plane angles of figures of three different kinds, one kind having an odd number of sides, cannot come together in a perfect solid figure.

For, again by 27, the solid angles would not all be the same, which is not in accordance with the definitions.

Part 25: Proposition

The most perfect regular congruences of plane figures to form a solid figure are 5 in number.

This is a scholium to the last proposition of the last book of Euclid.

Proposition 15 says that we must start with 3 plane angles

Proposition 16 says we must finish at:

• 6 trigon angles
• 4 tetragon angles
• 3 hexagon angles

This is because Proposition 18 says they add up to 4 right angles.

3e trigons, fitted together at one angle, make up less than 4 right angles in the plane, in fact they make up only two. When we make a solid angle by putting three trigons together the gap which remains can be filled by a fourth trigon.

This gives the Tetrahedron or Pyramid.

4 trigons, fitted together at one angle, make up eight thirds of a right angle, which is less than twelve thirds, or four right angles. Joining together the sides of the trigons we obtain a pyramid with an open four-sided base.

Two such pyramids may be fitted together base to base to form a figure closed on all sides. This gives the Octahedron.

5 trigons, fitted together at one angle, make up 10 thirds of a right angle, which is less than 12 thirds.

Joining the sides together two by two round the common angle we obtain a pyramid with a five-sided base. Each of the angles around the base must eventually be made up of five plane angles, so each requires another three angles in addition to the two it already has.

Thus the 10 plane angles around the base require another 15. 15 angles will then point outwards in the opposite direction which adds up to 30 plane angles, that is the angles of 10 trigons.

These 10 trigons make up a central zone or column, with a five-sided open end at top and bottom. Another penta­ hedral pyramid fits onto the open base, thus closing the figure all round. This gives the Icosahedron.

We have now dealt with all cases involving only trigons.

3 tetragon angles are three right angles, less than four right angles in the plane. Therefore they can be fitted together to form a solid angle.

When the tetragons are fitted together, they leave three gaps, and 3 angles of the plane stick out. So three more tetragons, fitted together to form a solid angle, will fit together with the first three, their points filling the gaps in the others and their gaps taking the points of the others. This gives the Hexahedron or Cube.

4 tetragon angles are 4 right angles, therefore by Part 16 they do not form a solid angle. So we have dealt with all cases forming only tetragons.

3 pentagon angles are 18 fifths of a plane right angle, which is less than 20 fifths, or 4 right angles.

Therefore, they can be fitted together to form a solid angle. If we take one pentagon as a base and fit five others round it in this manner, we obtain a figure with gaps which are five pentagon angles and points which stick out which are also five plane pentagon angles.

We can then construct another similar figure, in reverse, so that the five plane angles sticking out of the second figure will fit into the five gaps of the first one, and vice versa. This produces the Dodecahedron.