reach out and touch space - dartmouth collegemutzel/reachout_touchspace.pdf · reach out and touch...
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![Page 1: Reach out and touch space - Dartmouth Collegemutzel/reachout_touchspace.pdf · Reach out and touch space Author: Bjoern J. Muetzel Created Date: 12/12/2018 3:34:33 PM](https://reader033.vdocuments.net/reader033/viewer/2022051607/6035d0708b52977c1c1d1eb0/html5/thumbnails/1.jpg)
REACH OUT AND TOUCH SPACE
Bjoern Muetzel (Dartmouth College) - Family Night - MoMath
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SPECIAL THANKS TO
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REACH OUT AND TOUCH SPACE
![Page 4: Reach out and touch space - Dartmouth Collegemutzel/reachout_touchspace.pdf · Reach out and touch space Author: Bjoern J. Muetzel Created Date: 12/12/2018 3:34:33 PM](https://reader033.vdocuments.net/reader033/viewer/2022051607/6035d0708b52977c1c1d1eb0/html5/thumbnails/4.jpg)
SUPPOSE YOU WANT TO TILE YOUR
BATHROOM. WHICH REGULAR SHAPES OR
POLYGONS COULD YOU USE?
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Suppose you want to tile your bathroom. Which regular shapes or polygons could you use?
![Page 6: Reach out and touch space - Dartmouth Collegemutzel/reachout_touchspace.pdf · Reach out and touch space Author: Bjoern J. Muetzel Created Date: 12/12/2018 3:34:33 PM](https://reader033.vdocuments.net/reader033/viewer/2022051607/6035d0708b52977c1c1d1eb0/html5/thumbnails/6.jpg)
Suppose you want to tile your bathroom. Which regular shapes could you use?
![Page 7: Reach out and touch space - Dartmouth Collegemutzel/reachout_touchspace.pdf · Reach out and touch space Author: Bjoern J. Muetzel Created Date: 12/12/2018 3:34:33 PM](https://reader033.vdocuments.net/reader033/viewer/2022051607/6035d0708b52977c1c1d1eb0/html5/thumbnails/7.jpg)
Suppose you want to tile your bathroom. Which regular shapes could you use?
![Page 8: Reach out and touch space - Dartmouth Collegemutzel/reachout_touchspace.pdf · Reach out and touch space Author: Bjoern J. Muetzel Created Date: 12/12/2018 3:34:33 PM](https://reader033.vdocuments.net/reader033/viewer/2022051607/6035d0708b52977c1c1d1eb0/html5/thumbnails/8.jpg)
Suppose you want to tile your bathroom. Which regular shapes could you use?
![Page 9: Reach out and touch space - Dartmouth Collegemutzel/reachout_touchspace.pdf · Reach out and touch space Author: Bjoern J. Muetzel Created Date: 12/12/2018 3:34:33 PM](https://reader033.vdocuments.net/reader033/viewer/2022051607/6035d0708b52977c1c1d1eb0/html5/thumbnails/9.jpg)
CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.
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CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.
1
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CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.
2
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CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.
3
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CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.
4
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CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.
1
2 43 5
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Suppose you want to tile your bathroom. Which regular shapes could you use?
Regular pentagons do not work,
but what about irregular ones?
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CAN WE BE A BIT MORE CREATIVE?
IDEA 2: USE IRREGULAR
PENTAGONS.
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CAN WE BE A BIT MORE CREATIVE?IDEA 2: USE IRREGULAR PENTAGONS
1
2 43 5
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THERE ARE 15 FAMILIES OF
PENTAGON TILINGS, THE LAST ONE WAS
DISCOVERED IN 2015.
Pentagonal tilings
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Now we want to make symmetric polyhedra using these regular polygons.
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Which one does not work?
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Which one does not work?This time it is the hexagon!
Platonic solids
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THE 5 PLATONIC SOLIDS
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DUALITYoctahedron/cube
dodecahedron/icosahedron
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WHAT ABOUT THE SPHERE?
THE SPHERE DOES NOT COUNT AS A PLATONIC SOLID.
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HOWEVER, WE CAN USE THE PLATONIC SOLIDS TO TILE THE
SPHERE!
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NOTE: ON A SPHERE THERE ARE NO STRAIGHT
LINES.
INSTEAD WE USETHE LINES THAT
MINIMIZE THE DISTANCE BETWEEN TWO POINTS.
THESE ARE THE GREAT CIRCLES AND ARE ALSO
CALLED GEODESICS.
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Spherical tilings coming from the platonic solids
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THERE ARE MANY
SPHERICAL TILINGS, ALSO WITH SEVERAL
POLYGONS.
spherical tilings
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THERE ARE MANY SPHERICAL
TILINGS, ALSO WITH SEVERAL
POLYGONS.
LET’S MAKE A CUBOCTAHEDRON.
spherical tilings
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Besides flat and spherical there is a third standard geometry. This is the hyperbolic geometry.
We do not see this one very often, as it does not fit well in our space.
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THIS HYPERBOLIC GEOMETRY CAN BE DESCRIBED AS A DISK, WHERE GEODESICS, OR SHORTEST PATHS ARE CIRCLES MEETING THE BOUNDARY AT
AN ANGLE OF 90 DEGREES OR STRAIGHT LINES PASSING THROUGH THE CENTER.
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ADDITIONALLY DISTANCES ARE MEASURED DIFFERENTLY, SUCH THAT ALL HEXAGONS ON THE LEFT HAND SIDE HAVE THE SAME SIZE.
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THIS SPACE ALLOWS FOR THE MOST VARIETY OF TESSELLATIONS.
hyperbolic_tilings
(courtesy of Malin Christersson)
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IT INSPIRED M.C. ESCHER FOR HIS FAMOUS PICTURES.
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THESE CAN NOT BE USED TO TILE THE BATHROOM, BUT MAKE GREAT PLATES FOR THE KITCHEN!
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THANK YOU FOR YOUR ATTENTION