Geometry Notes
N-Dimensional Solids
Dimensional solids,
| Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ... |
| Number of faces | 0 | 1 | 6 | 12 | 20 | 30 | 42 | 56 | 72 | 90 | ... |
The formula for term index n, n starting at 2:
an = (n−1)n for n≥2.
Interesting
Creating a line segment
in 1699 with parchment, ruler, ink, and quill pen.
Be cool to have the above
Deutsche Fotothek
image(Geometry & Construction & Distance & Measuring Instrument)
beside an image of me creating a line segment with laptop and LibreOffice Draw.
Geometric Definitions
Regular polygon
is a polygon with equal length edges, equal angles, and convex .
Examples, an equilateral triangle is a regular triangle which has equal sides and equal angles.
A square is a regular quadrilateral.
A rhombus quadrilateral is not a square, then it is not a regular quadrilateral
because it's angles are not all equal even though it has equal length sides.
Convex polygon
means that the line segment between two points of the polygon
is contained in the boundary(inside) of the polygon.
An example of a convex polygon: a regular pentagon.
In 1619 Kepler defined stellation for polygons as the process of extending edges
until they meet to form a new polygon.
Stellation
is the process of extending a polygon in two dimensions, a polyhedron in three dimensions,
or, in general, a polytope in n dimensions to form a new figure.
"Stella" in Latin, is "star".
A compound to 2 squares where one is rotated 45o from the other,
creates a star like shape, a stellation polygon.
In Johannes Kepler's Harmonices Mundi (1619, The Harmony of the World),
the rhombic dodecahedron is one of his "rhombic solids" which is a polyhedra with all faces being congruent rhombuses.
The dodecahedron is composed of 12 faces and is related to the cube and octahedron
From this PDF. The five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Coxeter became a prominent mathematician, well-known for his work on geometry and symmetry, for example studying and classifying symmetries of higher dimensional figures. Coxeter and Roger Penrose met Escher for the first time, at an exhibition of Escher's work. and he bought a couple Penrose came up with ‘Penrose triangles’ after seeing Escher’s Relativity impossible staircase print. Infinity in a finite picture: hyperbolic tiling from Coxeter’s paper and Circle Limit I, M.C. Escher (1958).