By Irving Adler
More than a hundred workouts with solutions and two hundred diagrams remove darkness from the textual content. academics, scholars (particularly these majoring in arithmetic education), and mathematically minded readers will have fun with this remarkable exploration of the position of geometry within the improvement of Western medical thought.
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Additional info for A New Look at Geometry
X·(y·z) = (x·y)·z. 6. x·y = y·x. 7. There exists a member 1 such that, for every x in the system, x·1 = 1·x = x. 8. For each x there exists a member denoted by that has the property A comparison of the properties of + in F with those of · in F* shows a great similarity between them. Condition 5 is like condition 1, except that · takes the place of + in it. Similarly, condition 6 is like condition 2. Condition 7 is like condition 3, with the number 1 playing the same role with respect to multiplication that the number 0 plays with respect to addition.
Consequently, when we finally obtain the network of edges formed by the simple polyhedron, the value of V – E + F is the same as its initial value 2. But F now is the number of faces of the simple polyhedron. This proves that in a simple polyhedron the number of vertices V, the number of edges E, and the number of faces F are related by the formula The values of V, E, and F for each of the regular polyhedra are shown in the table below. This table reveals an interesting symmetry relationship that connects each regular polyhedron with another.
Therefore there are only three ways of filling a plane by repeating a regular n-gon: use equal regular 6-gons (regular hexagons) with three at each vertex, or equal regular 4-gons (squares) with four at each vertex, or equal regular 3-gons (equilateral triangles) with six at each vertex. The Regular Solids The three-dimensional analogue of a regular polygon is a regular solid. A regular solid is a polyhedron whose faces are congruent regular polygons arranged so that the same number of faces occurs at each vertex.
A New Look at Geometry by Irving Adler