If one draws all five internal diagonals of a
regular pentagon, the central region of the resulting five-pointed star in Figure 1 is a reduced mirror image of the initial pentagon formed by connecting its five points as a perimeter.
An affine
regular pentagon is uniquely determined by any three of its vertices.
Boselie explains that in this respect there is something wrong with the form of the
regular pentagon. This form is aesthetically inferior to the equilateral triangle, the square and the circle.
Now ask your students to determine the size of the internal angles of a
regular pentagon between any two adjacent sides.
A number of the planes of the crystal were
regular pentagons. This discovery defied all the laws of crystallography, which dictated that crystalline grids must be periodical: it is always possible to find a repeating unity in the grid, called the unity cell.
And the problem which both Caris and the crystallographers have in trying to explain the 'dimensional unruliness' of the
regular pentagon ultimately illustrates the very similarity between them in this context.
Other math examples include making your own unique pattern-tile shapes (like
regular pentagons) that are not part of normal kits but are very interesting to explore.
The cages have faces of 12
regular pentagons and up to 480 irregular hexagons, which puts them into a category of shapes called fullerenes.
Regular pentagons and hexagons have five and six equal sides and angles respectively.
The faces can be bounded by equilateral triangles, as are the tetrahedron, octahedron, and icosahedron; by squares, resulting in the cube; or by
regular pentagons, resulting in the dodecahedron.
There are precisely five such objects: the tetrahedron (made up of four equilateral triangles), cube (six squares), octahedron (eight equilateral triangles), dodecahedron (12
regular pentagons), and icosahedron (20 equilateral triangles).
(1) Similarly (and not surprisingly) the Dutch artist Gerard Caris always carries a small pentagonal dodecahedron made up of twelve
regular pentagons with him, so that he is always prepared to demonstrate the potentially infinite symmetrical mega-structures which can be made by combining
regular pentagons and the solids constructed from them.