Plane Congruence

Part 12: Congruent Plane Shapes

This is when they either enclose a solid figure or fill the plane without leaving a gap, the figures themselves being regular or semiregular.

Part 13: Incongruent

These regular plane shapes inscribed in a circle (if they can be inscribed) which cannot form either alone or with other plane shapes of their own or another class, a solid figure, other than a somewhat imperfect one, which can be inscribed in a spherical surface.

They are also unable to cover the plane, either by themselves or with stars of their own class, or with figures and stars of another class around them.

I have excluded the heptagon and suchlike figures, despite the fact that 2 parallel heptagons together with 7 square or 14 regular triangles do form a completely closed solid figure, because only two heptagons are involved.

The shape formed is discus-shaped, like a plane, not globe-shaped, like a sphere. See the figures marked A and B in the engraved plate following page

The 15-sided figure is also excluded in the same way, despite the fact that some of its angles may be surrounded by related figures to cover the plane, because in this case the figure is not completely surrounded at all its angles.

Proposition 14: At least three plane angles are required to form a congruence in the plane

Around any meeting-point the sum of the angles is four right angles.

But no figure has an angle greater than two right angles, therefore two such angles are less than four right angles. So two of them cannot fill the plane, by Definition 1.

Proposition 15: At least three plane angles must fit together or rise up to form a solid angle

For two plane angles would meet not only at their sides but with their whole surfaces, which is contrary to Euclid’s definition of a solid angle.

Proposition 16: The sum of angles congruent in the plane is always four right angles, never more

The sum of angles which form a solid congruence is less than 4 right angles.

For in a plane there are no more than four right angles around a point, therefore when the sum of the angles is equal to four right angles no gap is left, and by Definition I there is then congruence in the plane.

If the angles cover the plane they do not rise from it to form a solid angle.

On the other hand, if the angles fitted together in the plane leave a gap, that is if they come to less than four right angles, then drawing together the two sides round the gap, and so eliminating it, necessitates raising the angle and making it a solid one.

Figure H in the engraved plate, following page 53,’-’ shows three pentagons lying in the plane and leaving a gap.

Proposition 17: A figure with an odd number of sides, around which figures of 2 kinds are fitted, cannot form a congruence which is the same at every angle, either in the plane or in space

For one angle of the figure will have the same figure on both sides of it, which is not the case for the other angles. The reason for this can be seen in figure C of the engraved plate below.

Proposition 18: There are only 3 ways in which the plane can be filled most perfectly around a point.

Each case uses shapes of only 1 kind, by using:

• 6 trigons
• 4 tetragons or
• 3 hexagons.

Book 1 Part 33 said that the angle of a trigon is 2/3 of a right angle. Therefore the 6 angles of 6 trigons are 12 thirds, D.

that is four whole right angles. See D.^^

Similarly, the angle of a tetragon is one right angle. Therefore, the 4 angles of 4 tetragons make 4 right angles.

See E.

Similarly, the angle of a hexagon E is 8 sixths of a right angle. Therefore 3 angles of 3 figures make 24 sixths, that is 4 right angles.

See E

But the angle of a pentagon is less than that of a hexagon. Therefore, 3 of them are less than 4 right angles and leave a gap.

The angle of a pentagon is larger than that of a tetragon, therefore four pentagon angles are more than four right angles, therefore they cannot be contained around a point in a plane, by Part 16 of this book.

For this see H, where the fourth pentagon is shown dotted.

Similarly, the angles of a heptagon and of all larger figures are greater than that of a hexagon, so three heptagon angles are more than four right angles. See I, where two of the heptagons partly overlap in the plane.

Here we must consider rhombi made up of two regular trigons.

They form a most perfect congruence, like regular hexagons, although they are semiregular figures. This congruence can be seen in the engraved plate, labelled G.

Here we must also consider the six-cornered stars we obtain by removing 6 points from a star dodecagon. See letter K.

For where we have removed a point we have a re-entrant angle, equal to a right angle. Therefore three tet­ ragons and three points of these stars fill the plane.

For the hexagon can be divided up into one such star and six half-tetragons.