04/20/23

Polygons• Polygons are simple closed plane figures made with three or more line segments.

•Polygons cannot be made with any curves.

•Polygons are named according to their number of line segments, or sides.

 A regular polygon is a polygon with all sides congruent and all angles congruent such as equilateral triangle, square, regular pentagon, regular hexagon, ………..

04/20/23

Regular PolygonsName number of sides Angle at vertex

triangle 3 60quadrilateral 4 90pentagon 5 108hexagon 6 120heptagon 7 ~128.6octagon 8 135nonagon 9 140decagon 10 14411-gon 11 ~147.3dodecagon 12 150n-gon n {(n-2)180}/n

  A polyhedron is called a regular polyhedron

or Platonic Solids if the faces of the polyhedron are congruent regular polygonal regions, and if each vertex is the intersection

of the same number of edges.

Polyhedra is the plural for polyhedron.

Relationship Between Polygons and PolyhedronsA polyhedron and polygon share some of

the same qualities. A regular polyhedron’s face is the shape of a regular polygon.

For example: A tetrahedron has a face that is an equilateral triangle. This means that every face that makes the tetrahedron is an equilateral triangle. Around all the vertices and every edge is the same equilateral triangle.

Relationship Between Polygons and Polyhedrons

A polyhedron is made of a net which is basically like a layout plan. It is flat and made of all the faces that you will see on the polyhedron.

For example: A cube has six faces all of them are squares. When you open the cube up and lay it out flat you see all of the six squares that make up the cube.

Platonic Solids

Tetrahedron

Platonic Solids

Octahedron

Tetrahedron

Platonic Solids

Octahedron

Tetrahedron

Icosahedron

Platonic Solids

Cube

Octahedron

Tetrahedron

Icosahedron

Platonic Solids

Hexahedron

Octahedron

Dodecahedron

Tetrahedron

Icosahedron

~~There are only There are only fivefive platonic platonic solidssolids~~

Five “Regular” Polyhedra

Platonic solids were known to humans much earlier than the time of Plato. There are carved stones (dated approximately 2000 BC) that have been discovered in Scotland. Some of them are carved with lines corresponding to the edges of regular polyhedra.

Icosahedral dice were used by

the ancient Egyptians.

Ancient Roman Dice

ivory

stone

Evidence shows that Pythagoreans knew about the regular solids of cube, tetrahedron, and dodecahedron.

A later Greek mathematician, Theatetus (415 - 369 BC) has been credited for developing a general theory of regular polyhedra and adding the octahedron and icosahedron to solids that were known earlier.

Nets of Platonic Solids: Nets of Platonic Solids: http://agutie.homestead.com/files/solid/plathttp://agutie.homestead.com/files/solid/platonic_solid_1.htmonic_solid_1.htm

The name “Platonic solids” for regular polyhedra comes from the Greek philosopher Plato (427 - 347 BC) who associated them with the “elements” and the cosmos in his book Timaeus.

“Elements,” in ancient beliefs, were the four objects that constructed the physical world; these elements are fire, air, earth, and water. Plato suggested that the geometric forms of the smallest particles of these elements are regular polyhedra.

Fire is represented by the tetrahedron, air the octahedron, water the icosahedron, cube the earth, and the almost-spherical dodecahedron the universe.

 

Harmonices Mundi

Johannes Kepler

Symbolism from Plato:Octahedron = air

Tetrahedron = fire

Cube = earth

Icosahedron = water

Dodecahedron =

the universe

We define the dual of a regular polyhedron to be another regular polyhedron, which is formed by connecting the centers of the faces of the original polyhedron

Among the most important of M.C. Escher's works from a mathematical point of view are those dealing with the nature of space itself.

His woodcut Three Intersecting Planes is wonderful example of his work with space because it exemplifies the artist’s concern with the dimensionality of space, and with the mind's ability to discern three-dimensionality in a two-dimensional representation.

Escher’s Work with Polygons Escher’s Work with Polygons and Polyhedronsand Polyhedrons

M.C. Escher (1898-1972)

Stars, 1948

M.C. Escher

Double Planetoid, 1949

M.C. Escher

Waterfall, 1961

M.C. Escher

Reptiles, 1943

Cube with Ribbons

(lithograph, 1957)

Euler’s Formula and Platonic SolidsEuler’s Formula and Platonic Solids

http://www.mathsisfun.com/geometry/platonic-solids-why-five.html

  Faces Vertices Edges

Tetrahedron 4 triangular    

Hexahedron (Cube) 6 square    

Octahedron 8 triangular    

Dodecahedron 12 pentagonal    

Icosahedron 20 triangular    

,

Leonhard Euler • Euler = “Oiler”

See the fabulous TOOL for investigating Platonic Solids http://illuminations.nctm.org/imath/3-5/GeometricSolids/GeoSolids2.html

Platonic Solids Rock Videohttp://www.teachertube.com/viewVideo.php?video_id=79050

  Platonic Solids on Wikipediahttp://en.wikipedia.org/wiki/Platonic_solid

  Platonic Solids - Wolfram Math Worldhttp://mathworld.wolfram.com/PlatonicSolid.html Math is Fun Platonic Solidshttp://www.mathsisfun.com/platonic_solids.htmlInteractive Models of Platonic and Archimedean Solids http://www.scienceu.com/geometry/facts/solids/handson.html  

Mathematics Enclyclopedia http://www.mathacademy.com/pr/prime/articles/platsol/index.asp Platonic Solids http://www.math.utah.edu/~alfeld/math/polyhedra/polyhedra.html  The Mathematical Art of M.C. Escher http://www.mathacademy.com/pr/minitext/escher/ Polyhedra http://www.zoomschool.com/math/geometry/solids/  Platonic Solids and Plato's Theory of Everything http://www.mathpages.com/home/kmath096.htm  The Geometry Junkyard http://www.ics.uci.edu/~eppstein/junkyard/polymodel.html