The 5 Elements

Proposition 27

Most perfect solid congruences are also formed by semiregular figures, that is plane rhombi. There are only 2 cases.

From 12 plane rhombi whose diagonals are in a particular ratw we may make a solid rhombus which has the shape of a honeycomb cell, that is, it has six sides and an end in the form of a trihedral angle.

If 6 rhombi are fitted together in such a way that obtuse angles meet obtuse angles and acute angles meet acute angles, then there will be 3 obtuse-angled gaps and acute-angled projections above. The same will happen below.

So 3 rhombi may be fitted in both above and below, with their obtuse angles fitted together to make a solid point and their remaining angles going into the gaps in the first figure, while the gaps left between the three new rhombi take the projecting points of the original figure.

Similarly, 30 plane rhombi, with a different ratio between their diagonals, make a solid triacontahedral rhombus. The rhombi are joined together at their acute angles, 5 by 5, to give 2 solid angles pointing in opposite directions.

There are gaps left where the obtuse angles meet. Each set of five gaps is then filled by the obtuse angles of a further 5 rhombi. Between these 2 shell-like figures, we introduce a zone made of 10 rhombi joined together.

This is then joined to each shell.

We can show as follows that there are no further perfect congruences of rhombi. Two of the angles of a plane rhombus are acute and two obtuse, the sum of one acute angle and one obtuse one being two right angles.

Further, it is not possible to put together more than three obtuse angles, since their sum would be greater than four right angles. By joining up only three acute angles one obtains something like a cube, a rhombic hexahedron, which has only two acute solid angles, the pair furthest away from one another.

The other solid angles, in the middle of the body, lie closer together. The body does not satisfy Part 8, which does not admit cases where only two solid angles lie on the same sphere. Moreover, each of the six obtuse solid angles is formed by two obtuse plane angles and one acute one, an irregularity which is once more con­ trary to the definitions. Therefore we may not fit together only three acute plane angles.

But six angles, of six rhombi, will not fit together either. For if the individual acute angles are each two thirds of a right angle, the obtuse angles will be twice that size, that is four thirds. Thus both three obtuse angles and six acute angles will add up to four right angles, and neither the one set nor the other will form a solid angle, but instead the rhombi will cover the plane continuously, as in G.

If we now take smaller acute angles the corresponding obtuse angles will be larger than before and three of them will add up to more than 4 right angles.

Therefore there are only two most perfect congruences of rhombi: one in which four acute angles of the rhombi make up a solid angle and another in which five do. However, the cube might be added to the list, as the first rhombic solid, for its faces also have four equal sides, as do those of the solid rhombi.