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REACH OUT AND TOUCH SPACE Bjoern Muetzel (Dartmouth College) - Family Night - MoMath
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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 ...

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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 ...

REACH OUT AND TOUCH SPACE

Bjoern Muetzel (Dartmouth College) - Family Night - MoMath

Page 2: 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 ...

SPECIAL THANKS TO

Page 3: 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 ...

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 ...

SUPPOSE YOU WANT TO TILE YOUR

BATHROOM. WHICH REGULAR SHAPES OR

POLYGONS COULD YOU USE?

Page 5: 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 ...

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 ...

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 ...

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 ...

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 ...

CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.

Page 10: 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 ...

CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.

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Page 11: 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 ...

CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.

2

Page 12: 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 ...

CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.

3

Page 13: 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 ...

CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.

4

Page 14: 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 ...

CAN WE BE A BIT MORE CREATIVE?IDEA 1: USE SEVERAL DIFFERENT SHAPES.

1

2 43 5

Page 15: 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 ...

Suppose you want to tile your bathroom. Which regular shapes could you use?

Regular pentagons do not work,

but what about irregular ones?

Page 16: 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 ...

CAN WE BE A BIT MORE CREATIVE?

IDEA 2: USE IRREGULAR

PENTAGONS.

Page 17: 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 ...

CAN WE BE A BIT MORE CREATIVE?IDEA 2: USE IRREGULAR PENTAGONS

1

2 43 5

Page 18: 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 ...

THERE ARE 15 FAMILIES OF

PENTAGON TILINGS, THE LAST ONE WAS

DISCOVERED IN 2015.

Pentagonal tilings

Page 19: 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 ...

Now we want to make symmetric polyhedra using these regular polygons.

Page 20: 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 ...

Which one does not work?

Page 21: 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 ...

Which one does not work?This time it is the hexagon!

Platonic solids

Page 22: 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 ...

THE 5 PLATONIC SOLIDS

Page 23: 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 ...

DUALITYoctahedron/cube

dodecahedron/icosahedron

Page 24: 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 ...

WHAT ABOUT THE SPHERE?

THE SPHERE DOES NOT COUNT AS A PLATONIC SOLID.

Page 25: 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 ...

HOWEVER, WE CAN USE THE PLATONIC SOLIDS TO TILE THE

SPHERE!

Page 26: 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 ...

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.

Page 27: 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 ...

Spherical tilings coming from the platonic solids

Page 28: 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 ...

THERE ARE MANY

SPHERICAL TILINGS, ALSO WITH SEVERAL

POLYGONS.

spherical tilings

Page 29: 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 ...

THERE ARE MANY SPHERICAL

TILINGS, ALSO WITH SEVERAL

POLYGONS.

LET’S MAKE A CUBOCTAHEDRON.

spherical tilings

Page 30: 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 ...

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.

Page 31: 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 ...

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.

Page 32: 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 ...

ADDITIONALLY DISTANCES ARE MEASURED DIFFERENTLY, SUCH THAT ALL HEXAGONS ON THE LEFT HAND SIDE HAVE THE SAME SIZE.

Page 33: 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 ...

THIS SPACE ALLOWS FOR THE MOST VARIETY OF TESSELLATIONS.

hyperbolic_tilings

(courtesy of Malin Christersson)

Page 34: 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 ...

IT INSPIRED M.C. ESCHER FOR HIS FAMOUS PICTURES.

Page 35: 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 ...

THESE CAN NOT BE USED TO TILE THE BATHROOM, BUT MAKE GREAT PLATES FOR THE KITCHEN!

Page 36: 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 ...

THANK YOU FOR YOUR ATTENTION