The 5 Elements

Proposition 26

Ss and below, Book V, Ch. I. Tt in the previous plate.

We may add to the most perfect regular congruences two further ones, each involving twelve star pentagons, and two semisolid congruences, of star octagons and star decagons.

For star penta^ns form solid figures closed on all sides and having spikes. One figure has twelve quinquelinear angles and the other has twenty trilinear angles.

The former figure will stand up on three of its spikes, the latter on five at a time. The former looks handsomer if it stands upright on one spike, the latter sits more correctly when resting on five.

The outsides of these solids do not show a regular face but instead an isosceles triangle containing a pentagon angle. However, five such triangles always lie in a plane which has a five-cornered part covered by the body of the solid.

The triangles surround this pentagon as if it were their heart, and together with it they make up the star pentagon, a figure called Witch’s Foot in German, and by Paracelsus the sign of health.

In structure, this body resembles its faces: in the face, a star pentagon, the .sides of two triangles always lie in a straight line whose interior part not only forms the base of aii exterior triangle but at the same time is the side of an inner 5-cornered figure.

Similarly, in the solid, individual isosceles triangles from 5 solid angles lie in a plane and the five-cornered innermost marrow or heart of the five triangles, or of the star, either forms the base of one of the protruding solid angles, or, in the other solid, is the base for five solid angles.

These shapes are so closely related the one to the dodecahedron and the other to the icosahedron that the latter two figures, particularly the dodecahedron, seem somehow truncated or maimed when compared to the figures with spikes. The sides of the first and fourth points of star octagons and star decagons lie in a line, which passes through two intermediate points, and the stars can be fitted together with such sides joined two by two. The star octagons make a kind of cube, and the star decagons a kind of dodecahedron, figures which have not angles but ears, for when two of the plane angles are fitted together they must leave a gap, which cannot be closed.

Therefore by Part 11 the congruence is only semisolid.

These solid and semisolid congruences are called most perfect because as solids they fit definition VI of this book. Their faces fit the definition of a perfect figure, which is the second definition of Book I, that is, they are secondary perfect figures. Nor is it absurd to call a semisolid congruence most perfect, because what we are concerned with is a congruence to which definition VI would apply if it could be completed, though definitions IX and X would not.