Regular polyhedrons were known in Ancient Greece. They carry the name of "plane-tree" bodies. Four regular polyhedrons – a tetrahedron, an icosahedron, a cube and an octahedron – represent four "essence", elements. The octahedron symbolizes air.
It is required to you
- - paper;
- - pencil;
- - ruler.
Instruction
1. The octahedron consists of eight sides which are the correct triangles. All parties are equal in the correct triangle among themselves. Corners between the parties of such triangle make 60 °. Heights, medians, bisectors coincide. Creation of the correct octahedron requires a cube.
2. To construct a cube, draw a square. Recede some distance to the right and up, construct one more same square (the left and lower lines will be dashed). Connect the corresponding pair points of both squares before visualization of a cube. As on its basis you will build an octahedron, make it big and accurate.
3. Let the cube be given. It is necessary to construct the octahedron entered in it. Carry out diagonals for each side of a cube. Note points of intersection of diagonals. Connect all received points with each other. The correct octahedron entered in a cube is ready.
4. For the proof that the received figure – the correct octahedron, it is necessary to prove correctness of triangles. To prove that triangles sides – correct, carry out perpendiculars from their tops to cube edges. Use properties of rectangular triangles and a cube.
5. It is also possible to construct the octahedron described about the set cube. Let a be cube edge length. Find the centers of each side (these are points of intersection of diagonals). Draw straight lines through the centers of opposite sides. They will be crossed in the center of a cube which can be designated for a point of the Lake.
6. So, there are two straight lines which are crossed in O. Otlozhite's point on each of straight lines on both sides the piece equal 3a/2. Connect the ends of the pieces received by you. It will also be a framework of the correct octahedron described about a cube.