10. Square pyramid, tetrahedron & cone, isosceles, right-angled & equilateral triangles
If you want to finish off a square space or column uniformly, you let planes rise from a flat square at the same angle. The edges of these planes have then the same inclination with respect to the base and meet at a point. This body is called a square pyramid. It is the simplest pyramid which you can see among the graves of the ancient Egyptian kings and at the tops of obelisks and towers, on pylons, posts, nails, iron clamps, etc., in other words, on all objects, where prismatic bodies are to be finished off uniformly .
You can make a simple model of a square pyramid by cutting a square piece of paper, place a knitting needle vertically through the point of intersection of its diagonals and link its tip to the corners of the base by strings. In its fundamental position, one edge of the base is parallel to the blackboard and the base is horizontal. The length of the knitting needle is called the height, its end point the vertex and the strings the edges of the pyramid. A pyramid has four side faces and one base, five corners, four base edges and four side edges.
The side edges of the quadratic pyramid are equally long, the side faces are isosceles triangles, which you see on roofs and as ends of windows, etc.
The Greeks preferred in their architecture obtuse-angled, isosceles triangles, the Romanic style acute-angled, the Gothic style equilateral triangles. Today, right-angled triangles prevail.
The two equally long sides of isosceles triangles are called legs, the third side the base, the point of intersection of the legs the peak of the triangle. You can draw such a triangle easily with a compass and ruler, if you know the lengths of the legs and base. You draw the base of length b and draw circles with radius a about its end points; the point of intersection of the circles is the peak of the triangle which you link to the endpoints of the base.
Pattern of the quadratic pyramid
The construction of right-angled triangles allows you to draw the pattern of the square pyramid. Just imagine it to have been cut along its side edges and the sides folded into the plane of its base. You see that its pattern comprises a square and four isosceles triangles.
If you construct below the base of a pyramid another pyramid, you obtain a double-pyramid. If the edges of both these pyramids are equally long, you obtain a octahedron (a regular solid with eight equal faces).
If you draw from the tip of an isosceles triangle the perpendicular to the base, it is decomposed into two right-angled triangles. This perpendicular is called its height. In right-angled triangles, the base opposite the right angle is called hypotenuse.
You construct a right-angled triangle for given lengths a and b of the shorter sides as follows: With the aid of a set square, draw a right angle, mark the lengths of the two legs a and b, and link the end points.
An especially simple isosceles triangle is the equilateral triangle in which all sides have the same length, so that each side can serve as a base. You construct such a triangle with side length a as follows: Draw one segment of length a, adjust the compass with its pin at one end point to the other end point and draw circular arcs around both ends with this radius. Join the ends of the original segment to the intersection of the two arcs.
With four isosceles triangles you can compose an especially simple body which, when placed on one of its faces, has the shape of a three-sided pyramid. Each of its sides can serve as base. It is called a regular tetrahedron.
The tetrahedron has 6 edges, because each side has three edges, i.e., 3·4 = 12 and every edge is counted twice. Every side yields 3 corners, so that it has 3·4 = 12 corners, but 3 edges meet at every corner and every corner is counted three times, whence the tetrahedron has 4 corners.
You obtain the pattern of the tetrahedron by cutting along its edges and folding the sides into the plane of the base.
If you want to cover a space with a circular ground plan, so that it is equally inclined all around to the base, you obtain a circular cone. You can see it on towers, tents, funnels, sharpened pencils. posts, spikes, etc., that is every time when a circular body is sharpened to a point.
A circle is the base, the distance from the base to the peak the height and the curved surface the mantle of the cone. You discover that you can place a ruler along a cone's sides from the base to its peak and draw its mantle lines. You obtain the pattern of the cone by cutting along one of its mantle lines, spreading its mantle into a plane and flipping the base beside it. The mantle is a circular sector. The length of the arc of the sector equals the circumference of the base.
If you place two equal cones with their bases together, you obtain a double cone.