The Congruence Of Regular Figures

Part 1: Congruence

Congruence takes one form in the plane and another in space. In the plane, there is congruence when individual angles of several shapes coming together at a point in such a way that they leave no gap.

Part 2: Perfect Congruence

This is when the angles of the figure which come together do so in the same way at each meeting-point, so that these meeting-points are similar to one another and the pattern of meeting-points can be continued indefinitely.

Part 3: Most Perfect Congruence

This is when, in addition, the figures which come together in the plane are all of the same kind.

Part 4: Imperfect Congruence

This is when some larger figure is surrounded by similar meeting-points but the congruence cannot be continued indefinitely or can be so continued only by introducing meeting points of different kinds. The congruence is imperfect, and of lower degree, when the larger figure cannot be surrounded in such a way that similar meeting-points are formed at all its angles.

Part 5: Congruence in space and a solid figure

This is when the individual angles of several plane figures make up a solid angle, and regular or semi-regular figures are fitted together so as to leave no gap between the sides of the figures, which join up on the opposite side of the solid figure, or, if a gap is left, it is such that it can be filled by a figure of one of the kinds already employed, or, at least, by a regular figure.

Note that there is another form of congruence, not of plane figures to form a solid figure but of these solid figures among themselves, to fill space all round a point. There are only two figures which will form such congruences: the cube and the rhombic dodecahedron.

8 cube angles will meet at a point and fill space all round it. The rhombic figure has two types of angle: eight obtuse trilinear angles and six acute quadrilinear angles.

Four obtuse angles will join to fill space, and so will six acute ones. The result is like the way in which bees construct their honeycomb: the cells are contiguous and the end of each is sur­ rounded by three opposing ends while its sides are surrounded by the sides of six more cells.

Three more cells could be added at the other end to complete the figure, except that the entrances to the cells must remain open. We are not concerned here with this congruence of solid figures.

Part 6: The Most Perfect Congruence in Space

This is when the solid is formed. In addition, the plane figures which form the congruence are all the same shape.

Part 7: Completely regular solid

This is formed when the plane figures are regular.

All its angles then lie on the same spherical surface and are all similar to one another.

Part 8: Semiregular Solid

This is formed when the plane figures are semiregular (see Book 1, Part 3).

Its solid angles are then not all the same, but differ in the number of lines they contain, though the angles are not of more than two kinds, and neither are they distributed on more than two spherical surfaces, which are concentric.

The number of angles of each kind must be the same as the number of angles of one of the regular solid figures.

There is no reason why we should not call this congruence most perfect.

for its imperfection is in the faces and is not a consequence of its being solid but, rather, an accidental feature. So this semiregular congruence will equally be called most perfect.

Part 9: Low Degree Perfect Congruence

This is when the plane figures are regular and all the angles lie on the same spherical surface and are similar to one another, but the faces are of various kinds, though the number of each kind must be the same as the number of faces of one of the most perfect figures, that is, not less than four, which is the minimum number of planes to bound a solid figure.

Part 10: Imperfect congruence or figure

This is when other conditions remain the same but the larger plane figure does not occur more than once or twice.

The solid figure formed will either be more like a part of a figure than a whole one^ or it will be more like a plane figure than a solid, since any solid figure is bounded by at least four surfaces.

Such figures are shown in the plate, marked A and B, where the larger figure is a heptagon. These two classes ex­ tend indefinitely as the number of sides of the larger figure increases.

They each start with the trigon, which in class A gives us one of the most perfect regular congruences.

Proceeding to the tetragon we then obtain one of the most per­ fect congruences in class

All other congruences of these types are imperfect.

Part 11: Semisolid Congruence

This is when it does not satisfy all the conditions of Congruence in space, so that as the plane figures are fitted together the congruence does not completely Join up with itself but leaves a gap.

Semisolid Congruence applies to Solids (Part 6) and Completely Regular Solids (Part 7)