{"id":3745,"date":"2022-09-26T15:51:53","date_gmt":"2022-09-26T13:51:53","guid":{"rendered":"https:\/\/erlebnisland-mathematik.de\/?page_id=3745"},"modified":"2023-06-21T15:02:28","modified_gmt":"2023-06-21T13:02:28","slug":"advanced-text-polyhedral-crown-platonic-solids","status":"publish","type":"page","link":"https:\/\/erlebnisland-mathematik.de\/en\/advanced-text-polyhedral-crown-platonic-solids\/","title":{"rendered":"Advanced Text Polyhedral Crown (Platonic Solids)"},"content":{"rendered":"

[vc_row drowwidth=”sidebar-biest-default sidebar-biest”][vc_column][vc_column_text]<\/p>\n

Platonic bodies<\/h1>\n

Platonic<\/em> solids (or: ideal solids<\/em>, regular polyhedra<\/em> — “polyhedra<\/em>“) are convex<\/em> solids with the greatest possible regularity, named after the Greek philosopher Plato<\/em> (427–347 BC). (Here a body is called convex<\/em> if with two each of its points \"P\" and \"Q\" also all points on the connecting line \"\overline{PQ}\" belong to it).<\/p>\n

This (“greatest possible<\/em>“) regularity consists in the fact that for each of these solids all side faces are congruent<\/em> (“congruent”) to each other and that they meet in the same way in each corner.<\/p>\n

There are exactly five Platonic bodies:[\/vc_column_text][vc_single_image image=”2394″ img_size=”large” alignment=”center”]Figure 1: The Five Platonic Bodies[\/vc_single_image][vc_column_text]Specifically, these five bodies have the following properties:<\/p>\n\n\n\n\n\t\n\n\t\n\t\n\t\n\t\n\t
<\/th>Side faces<\/th>Number of faces<\/th>Number of corners<\/th>Number of edges<\/th>Number of faces in a corner<\/th>\n<\/tr>\n<\/thead>\n
Tetrahedron<\/td>equilateral triangle<\/td>4<\/td>4<\/td>6<\/td>3<\/td>\n<\/tr>\n
Cube (Hex- ameter)<\/td>squares<\/td>6<\/td>8<\/td>12<\/td>3<\/td>\n<\/tr>\n
Octahedron<\/td>equilateral triangle<\/td>8<\/td>6<\/td>12<\/td>4<\/td>\n<\/tr>\n
Dodecahedron<\/td>regular pentagons<\/td>12<\/td>20<\/td>30<\/td>3<\/td>\n<\/tr>\n
Icosahedron<\/td>equilateral triangles<\/td>20<\/td>12<\/td>30<\/td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n

Figure 1: The properties of the five Platonic solids<\/p>\n

[\/vc_column_text][vc_column_text]The Platonic Bodies have played a significant role in intellectual history from ancient Greece through the Middle Ages to our own time. Tetrahedron<\/em>, hexahedron<\/em> (cube) and dodecahedron<\/em> were well known to the students of Pythagoras in the 6th century BC. Theaitetos<\/em> (4th century BC) was also familiar with octahedron<\/em> and icosahedron<\/em>.<\/p>\n

Platon has described the solids, which were named after him later, in his work Timaisos<\/em> detailed and assigned to them the four elements, which were according to the opinion at this time the “World building blocks<\/em>“, in the following way:<\/p>\n