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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?