2. DEFINITION: TILES AND TILING
13.3
DEFINITION: TILES AND TILING
A simple closed curve, together with its interior, is a tile. A set of tiles forms a tiling of a figure if the figure is completely covered by the tiles without overlapping any interior points of the tiles.
In a tiling of a figure, there can be no gaps between tiles. Tilings are also known as tessellations.
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3. TILING WITH REGULAR POLYGONS
13.3
TILING WITH REGULAR POLYGONS
Any arrangement of nonoverlapping polygonal tiles surrounding a common vertex is called a vertex figure.
Equilateral triangles form a regular tiling because the measures of the interior angles meeting at a vertex figure add to 360.
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4. TILING WITH EQUILATERAL TRIANGLES
13.3
TILING WITH EQUILATERAL TRIANGLES
One interior angle of an equilateral triangle has measure 60.
At a vertex angle:
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5. One interior angle of a square has measure 90.
13.3
TILING WITH SQUARES
One interior angle of a square has measure 90.
At a vertex angle:
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6. TILING WITH REGULAR HEXAGONS
13.3
TILING WITH REGULAR HEXAGONS
One interior angle of a regular hexagon has measure
At a vertex angle:
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7. TILING WITH REGULAR PENTAGONS?
13.3
TILING WITH REGULAR PENTAGONS?
One interior angle of a regular pentagon has measure
At a vertex angle:
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8. THE REGULAR TILINGS OF THE PLANE
13.3
THE REGULAR TILINGS OF THE PLANE
There are exactly three regular tilings of the plane:
by equilateral triangles,
by squares, and
by regular hexagons.
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9. TILING THE PLANE WITH CONGRUENT POLYGONAL TILES
13.3
TILING THE PLANE WITH CONGRUENT POLYGONAL TILES
The plane can be tiled by:
any triangular tile;
any quadrilateral tile, convex or not;
certain pentagonal tiles (for example,
those with two parallel sides);
certain hexagonal tiles (for example,
those with two opposite parallel sides of
the same length).
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10. SEMIREGULAR TILINGS OF THE PLANE
13.3
SEMIREGULAR TILINGS OF THE PLANE
An edge-to-edge tiling of the plane with more than one type of regular polygon and with identical vertex figures is called a semiregular tiling.
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11. ITS GRID OF PARALLELOGRAMS
13.3
TILINGS OF ESCHER TYPE
Dutch artist Escher created a large number of artistic tilings.
ESCHER’S BIRDS
ITS GRID OF PARALLELOGRAMS
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12. MODIFYING A REGULAR HEXAGON WITH ROTATIONS
13.3
TILINGS OF ESCHER TYPE
MODIFYING A REGULAR HEXAGON WITH ROTATIONS
CREATES:
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