# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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To change °C to °F we use the formula F = |C + 32.

Example 1. Convert 37°C to the Fahrenheit scale.

Solution. Write

F = | x 37 + 32 Substituting 37 for C F = 98.6

37°C converts to 98.6°F.

To convert °F to °C we use the formula C = |(F — 32).

Example 2. Convert 212°F to the Celsius scale.

TESSELLATIONS 437

Solution. Write

C = | x (212 - 32) Substituting 212 for F C = 100

212°F converts to 100°C.

References: Conversion, Formula.

TERMINATING DECIMALS

Reference: Decimal.

TERMS

Reference: Algebra.

TESSELLATIONS

Tessellations are also known as mosaics. A tessellation is a tiling pattern that covers a flat surface. The tiles we often use are regular polygons such as equilateral triangles, squares, or regular hexagons, but a variety of shapes may be used. Some tessellations involve more than one shape, such as combining square tiles with regular octagons. A tessellation is only correct if the tiles fit together with no gaps and no overlapping and a definite repeated pattern can be easily recognized.

Tiling a kitchen floor with a square tile is a tessellation. Three examples are shown in figure a. If you know where to put the next tile, you probably have an easily recognizable pattern.

Tessellations occur in nature. A good example is the shape of the cells that form the honeycomb in a beehive. The tile is a regular hexagon (see figure b). Each interior angle of a regular hexagon is 120°. At the point where three hexagons meet they will completely “surround” the point with no gaps, because 3 x 120° = 360°, which is a full turn. To see how triangles, quadrilaterals, hexagons, and octagons tessellate, see

438 TESSELLATIONS

their respective entries. Some regular polygons do not tessellate, because when they meet at a point they do not surround the point in a way that leaves no gaps.

Example. Explain why regular pentagons do not tessellate.

Solution. Each interior angle of a regular pentagon is 108°. If three regular pentagons meet at a point, the angle sum at the point is 3 x 108° = 324°, which is less than a full turn. If four of them meet at a point, the angle sum is 4 x 108° = 432°, which is more than a full turn. The regular pentagon will not tessellate, because each interior angle is 108° and there are no combinations of this angle that will make 360°, so regular pentagons will not “surround” a point.

Two regular polygons that will combine to form a tessellation are the square and the octagon. Their tiling pattern is shown in figure d. If we examine how the shapes surround one point, say A, we can find out why these two shapes tessellate. Each interior angle of a regular octagon is 135° and each interior angle of a square is 90° (see figure d). At the point A, there are two octagons and one square. Therefore 135° + 135° + 90° = 360°, which is a full turn. A square and two octagons surround the point A.

Combinations of other regular shapes, in addition to the octagons and squares described above that tessellate, are listed here. The number of the shapes that surround a point is stated and their angle sum.

♦ Hexagon, two squares, one equilateral triangle (120° + 90° + 90° + 60° = 360°)

♦ One dodecagon, one square, two equilateral triangles (150° + 90° + 60° + 60° = 360°)

TETRAHEDRON 439

♦ Two squares, three equilateral triangles (90° + 90° + 60° + 60° + 60° = 360°)

♦ Two hexagons, two equilateral triangles (120° + 120° + 60° + 60° = 360°)

♦ One dodecagon, one hexagon, one square (150° + 120° + 90° = 360°)

♦ One hexagon, four equilateral triangles (120° + 60° + 60° + 60° + 60° = 360°)

It is interesting to tessellate irregular shapes. In figure e, the curved shape that is shaded produces the attractive tessellation of a queue of bald-headed men. The partly shaded tile, when tessellated, produces a set of steps.

The Dutch artist M. Escher constructed some extraordinary tessellations. In particular, his picture of knights on horseback is very striking.

References: Dodecagon, Equilateral Triangle, Hexagon, Mosaics, Octagon, Pentagon, Polygon, Quadrilateral, Regular Polygon, Square, Triangle.

TETRAHEDRON

This is a pyramid with four plane faces, each of which is a triangle. The prefix “tetra” means four. One of the four faces is called the base, and the vertex that is opposite to the base is called the apex (see figure a). The volume of a tetrahedron = | area of base x altitude.

Face

(a)

Example 1. Figure b shows a tetrahedron drawn inside a cube and occupying one of its comers. The apex of this tetrahedron is directly above the point which is one of

440 TETRAHEDRON

the vertices of its base. If the length of each edge of the cube is 5 cm, find the volume of the tetrahedron.

Apex

Solution. The area of the base of the cube is 5 x 5 = 25 cm2. The base of the tetrahedron is obviously half the area of the base of the cube and is 12.5 cm2'. The altitude of the tetrahedron is 5 cm. Write

Volume of tetrahedron = | x 12.5x5 Using the formula for the volume of

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