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

ISBN 0-471-15045-2

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Suppose you use a polo stick instead of a walking stick for the rotations (see figure d). The polo stick is rotated about the end of the handle each time, and the

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ORDER OF ROTATIONAL SYMMETRY 311

size of the angle between each polo stick is 90Â°. The complete figure has rotational symmetry of order four, and it has four axes of symmetry, which are drawn dashed in the right-hand side of figure d.

(d)

Some figures do not appear to have rotational symmetry. For example, consider three positions of the walking stick when the angles between the three sticks are 90Â°, 90Â°, and 180Â° (see figure e). In figures like this we say the figure does have rotational symmetry, but its order of rotational symmetry is one. Every figure has at least rotational symmetry of order one.

b

(e)

Summary 1. If a figure has n axes of symmetry, it will have rotational symmetry of order ft, except when n = 0, in which case the rule breaks down.

2. If a figure has rotational symmetry of order n, it will have n axes of symmetry or no axes of symmetry.

Example. Madge runs a courier business and the logo is a figure with three running legs (see figure f). What is the order of rotational symmetry of the logo?

312 ORTHOCENTER

Solution. There are three turns about the point at the center of the figure, and the turn of each leg is through the same angle of 120Â°. The logo has rotational symmetry of order three, but does not have any axes of symmetry.

Some playing cards have rotational symmetry of order two, as shown by the six of diamonds (see figure g). The center of rotational symmetry is at the center of the playing card. The card has no axes of symmetry. Not all playing cards have rotational symmetry of order two, because the pattern of the shapes has no symmetry, as in the seven of diamonds. Playing cards are designed with this symmetry so they can be easily read from opposite sides of a card table. Many crosswords are designed with a rotational symmetry order of four.

Total Order of Symmetry This is the result of adding the number of axes of symmetry that a shape has to its order of rotational symmetry. For example, the total order of symmetry of a square is 4 + 4 = 8. To find the symmetry of well-known polygons search for them under the relevant entry.

References: Axis of Symmetry, Center of Rotation.

ORDERED PAIRS

Reference: Cartesian Coordinates.

ORDINAL DATA

These are data that can be ranked according to size.

Reference: Data.

(9)

ORIGIN

Reference: Axes.

ORTHOCENTER

In the accompanying figure the three altitudes of the triangle meet at a point H called the orthocenter of the triangle. In the figure the altitudes of the triangle are AL, BM,

OVAL 313

and CN and their point of intersection is H. We say that the three altitudes of a triangle are concurrent, which means meet at a point.

/4

For a right-angled triangle ABC, with the right angle at the point C, the three altitudes of the triangle intersect at C. This means that the point C is the orthocenter of the triangle.

If the triangle is obtuse-angled, the orthocenter lies outside the triangle. References: Altitude Angle in a Semicircle, Collinear, Concurrent.

OUNCE

The ounce (abbreviation oz) is a unit of weight, and is an imperial unit. One ounce is part of a pound, which means 16 oz = 1 pound (lb). One ounce is approximately 28 grams (28.349532 grams, to 6 dp).

References: Gram, Imperial System of Units, Pound.

OUTCOME

Reference: Event.

OVAL

This is another name for ellipse.

Reference: Ellipse.

p

PALINDROMIC NUMBER

This is a number that reads the same backward as forward. For example, the number 1221 is a palindromic number, because when you read it backward you get the same number, 1221. Other examples of palindromic numbers are 12621, 22, and 4884.

Stringing together the last digit of each of the first nine square numbers forms a palindromic number:

l2 = l, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36,

72 = 49 , 82 = 64, 92 = 81

Writing down the last digit of each square number, we obtain the palindromic number 149656941.

Reference: Digit.

PARABOLA

Check the entry Quadratic Graphs for how to draw parabolas. Suppose you throw a stone through the air. Its height h meters after time t seconds may be modeled by the quadratic formula h = 201 â€” St2. The graph of this quadratic formula is the parabola shown in the figure. It was Galileo (1564-1643) who first discovered that when objects are thrown, their path through the air is in the shape of a parabola. This is only true when air resistance is neglected and gravity is uniform. The path of projectiles, fired from a gun or a bow and arrow, or of water coming out of a hosepipe, is a parabola.

One of the properties of the parabolic curve is made use of in car headlights. If a light bulb is placed at the focal point (F) of a parabolic reflector (see right-hand side of figure), the light rays will emerge from the headlight as parallel beams of light. This property is not true for other curves.

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