CREATING ART FROM MATHKAVITA RAMANAN AND ANNA GRIM

1

WHAT DO THESE HAVE IN COMMON?

KAVITA RAMANAN AND ANNA GRIM

2 31

WHAT DO THESE HAVE IN COMMON?

KAVITA RAMANAN AND ANNA GRIM

1 2 3

WHAT DO THESE HAVE IN COMMON?

THESE ARE ALL TILINGS!

A TILING IS A SET OF SHAPES THAT COVER A FLAT SURFACE WITHOUT ANY

GAPS OR OVERLAPS

WHAT IS A TILING?

IS THIS A TILING?

Surface

Tile

IS THIS A TILING?

BECAUSE THE TILES DON’T COMPLETELY COVER THE SURFACE

IS THIS A TILING?

NO!

IS THIS A TILING?

Surface

Tile

IS THIS A TILING?

IS THIS A TILING?

YES!1. COMPLETELY COVERS THE SURFACE

2. TILES DO NOT OVERLAP

IS THIS A TILING?

SurfaceTile

IS THIS A TILING?

BECAUSE THE TILES OVERLAP

IS THIS A TILING?

NO!

REGULAR VS. IRREGULAR TILINGS?

REGULAR IRREGULAR

PROBLEM 1CAN WE MAKE A TILING WITH ANY

REGULAR SHAPE?

PROBLEM 1CAN WE MAKE A TILING WITH ANY

REGULAR SHAPE?

PROBLEM 1CAN WE MAKE A TILING WITH ANY

REGULAR SHAPE?

PROBLEM 1CAN WE MAKE A TILING WITH ANY

REGULAR SHAPE?

PROBLEM 1CAN WE MAKE A TILING WITH ANY

REGULAR SHAPE?

PROBLEM 1CAN WE MAKE A TILING WITH ANY

REGULAR SHAPE?

SOLUTION 1WE CAN ONLY MAKE A TILING OUT OF

TRIANGLES, SQUARES, AND HEXAGONS

WHY?IT DEPENDS ON THE ANGLE OF

THE SHAPE

6∘

SMALL ANGLE = POINTY CORNER

90∘

BIG ANGLE = WIDE CORNER

120∘

LET’S TILE WITH SQUARES

LET’S TILE WITH SQUARES

90∘

90

LET’S TILE WITH SQUARES

9090+

LET’S TILE WITH SQUARES

180∘

9090+

180

LET’S TILE WITH SQUARES

9090+

18090+

LET’S TILE WITH SQUARES

270∘

9090+

18090+

270

LET’S TILE WITH SQUARES

9090+

18090+

27090+

LET’S TILE WITH SQUARES

360∘

9090+

18090+

27090+

360

LET’S TILE WITH HEXAGONS

120∘

120

LET’S TILE WITH HEXAGONS

+120120

LET’S TILE WITH HEXAGONS

+120120

240∘240

LET’S TILE WITH HEXAGONS

+120120240120+

LET’S TILE WITH HEXAGONS

+120120240120+360

360∘

LET’S TILE WITH PENTAGONS

108∘

108

LET’S TILE WITH PENTAGONS

108108+

LET’S TILE WITH PENTAGONS

216∘

108108+216

LET’S TILE WITH PENTAGONS

108108+216108+

LET’S TILE WITH PENTAGONS

324∘

108108+216108+324

COMPLEX TILINGS

COMPLEX TILINGS

COMPLEX TILINGS

MC. ESCHER

WHO MADE THOSE TILINGS?

ROGER PENROSE

PENROSE TILING

HOW T0 MAKE A COMPLEX TILING

ORIGINAL TILING DEFORMED TILING

RULE: CANNOT CHANGE SYMMETRY OF TILING

WHAT IS A SYMMETRY?WHEN A SHAPE LOOKS THE SAME AFTER MAKING A MOVE SUCH AS

ROTATION REFLECTION SHIFT

SYMMETRY IN TILINGSWHAT KINDS OF SYMMETRIES DO OUR

TILINGS HAVE?

WHAT SYMMETRIES DO YOU SEE?

WHAT SYMMETRIES DO YOU SEE?

WHAT SYMMETRIES DO YOU SEE?

BEYOND TILING FLAT SURFACES?

BEYOND TILING FLAT SURFACES?