STEM Cats Tessellations MC Escher Buckminsterfullerene Common Cold Virus Bogota Sport Center Epcot Dice Seven Days Sci Fi TV Series Rubies Giants Causeway Origami Lamps Projection of Cube ID: 544577
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Slide1
Polyhedra
STEM CatsSlide2
TessellationsSlide3Slide4Slide5Slide6Slide7Slide8
M.C. EscherSlide9
BuckminsterfullereneSlide10
Common Cold VirusSlide11
Bogota Sport CenterSlide12
EpcotSlide13
DiceSlide14
Seven Days
Sci
Fi
TV SeriesSlide15
RubiesSlide16
Giant’s CausewaySlide17
OrigamiSlide18
LampsSlide19
Projection of CubeSlide20
Projection of a HypercubeSlide21
Projection of a HypercubeSlide22
Many HypercubesSlide23
Delete Half of the Edges to get DiamondSlide24
Combinatorial Optimization
There are 100 people and 100 jobs. For each person and each job you know how much it will cost to (re)train that person to do that job. What is the least expensive matching of people to jobs?
The number of possible matchings is 100! (100 factorial). There is not enough time left in the life of the universe to run through this list to check the cost of each matching.
By understanding the properties of a certain polyhedron of dimension 10,000-199 (picture omitted!), this problem can be solved routinely in a fraction of a second.Slide25
Modeling, Visualization, and Math Art
Polyhedra
are basic building blocks for 2- and 3-d modeling.
Example of assembling a polyhedral puzzle, created with POV-Ray during my Visualizing Math course
:
http://www.ms.uky.edu/~lee/visualfa15/puzzle5.movSlide26
Challenge #1
Construct as many tessellations as you can with identical regular polygons Slide27
Challenge #2
Construct as many tessellations as you can with the following properties
All tiles are regular polygons
There are at least two different kinds of tiles
The same sequence of tiles meets around each vertexSlide28
Challenge #3
Construct as many
polyhedra
that you can with the following properties
All faces are equilateral triangles
All edges bend “outward”– the polyhedron is convexSlide29
Challenge #4
Construct as many
polyhedra
that you can with the following properties
All faces are identical regular polygons
The same number of faces meet at each vertex (corner)Slide30
Challenge #5
Construct as many
polyhedra
that you can with the following properties
All faces are regular polygons
There are at least two different kinds of faces
The same sequence of faces meets around each vertexSlide31
Challenge #6
Find
all flat shapes
formed by six squares that can be folded up to make a cube.
See
also
http://erikdemaine.org/metamorphosis
/Slide32
Image Sources
http://www.georgehart.com/virtual-polyhedra/escher.html
http://it.wikipedia.org/wiki/Fullereni#mediaviewer/File:Fullerene-C60.png
http://www.lightnessofbeingbook.com/LOBColorPlates/images/Plate03b_IcosVirus.jpg
http://www.domusweb.it/en/architecture/2012/01/04/el-bosque-de-la-esperanza.html
http://images.frpgames.org/products/product_36488.jpg
http://i1.sell.com/u/3H/5G/5BQ/4706433-l.jpg
http://images.nationalgeographic.com/wpf/media-live/photos/000/011/cache/rubies_1176_990x742.jpg
http://shearings-cdn2.zolvtravel.com/DynamicImages/d510/d510755c6e739c2465038ddea2f849d0_640_480-634873692367638135.jpg
http://michal.kosmulski.org/origami/icosidodecahedron-jitterbug.jpg
http://media.tumblr.com/tumblr_lefroeVII61qeb9gf.jpg
http://www.jeffreyventrella.com/Old_Art/cuboctahedron_big.jpg
http://www.wdwinfo.com/Photos/600-spaceship-earth3.jpg
http://
upload.wikimedia.org/wikipedia/commons/d/d9/Rhombic_dodecahedra.png