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Symmetry
Line Symmetry - A line of symmetry is the same as a line of reflection.
Rotational Symmetry - Figure 1 is said to have rotational symmetry because it fallsexactly upon top of itself before it is completely turned around a point. This figure hasrotation symmetry of 360/8 or 45°,90°, 135°, 180°,225°,270°,315°, and 360c
. Thiscan also be described as 1/8 turn symmetry. A
o0'
B
B'
A'
Figure 1 Figure 2
Point Symmetry - Figure 2 has point symmetry about point P because P is themidpoint of each segment drawn between a point and its image. A figure with pointsymmetry also has 1800 rotational or half-turn symmetry.
Find all lines of symmetry. then identify any rotational symmetries, andfigures with point symmetry,1. 2. 3.
4, 5. 6.
NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS SEPTEMBER 199:
Japanese family crest(stylized cherry blossom)
Rotocenter Revolution!
Tibetan poppy Decorative patternNocera, Italy
More examples follow (fig. 1). Ignore slight variations inpattern. since authentic designs rarely display perfectrotational symmetry.
~~
• 0t:Jj-~~~fr- ~~.~~~
: (a) (b) (c) (d)A,~O (~~e~~(e) (1) (g) (h)
••• ~( i) (j) (k) ( I)
Fig. 1
t. Which patterns have 1/2-turn rotational symmetry?
2. Which patterns have 1I3-turn rotational symmetry?
3. Which patterns have 1/4-turn rotational symmetry?
4. Using this same system, name the rotational symme-try tor sacn example. Note that an example can havemore than one kind of rotational symmetry.
(a) _(b)_(c)_(d) _(e)_(f)_
(g) _(h)_(i)_(j) _(k)_(I)_
A figure with 117-tum rotational symmetry is said to have7-fold rotational symmetry. If a figure has several tumsthat could describe its rotational symmetry, it is usual toname it using the smallest tractional tum possible. Exam-ple f has 112-tum, 1/4-tum, and 1/8-tum rotational sym-metry, so we say that it has 8-fold rotational symmetry.
One characteristic of any designwith rotational symmetry is thecenter point around which thedesign rotates. This point iscalled the rotocenter. A dot isplaced on the design at the rightto designate the rotocenter. Inthis example, the dot indicates a3-fold rotocenter.
5. Draw a dot to designate the rotocenter in each designin figure 1_
Some rotationally symmetricdesigns also have line symmetry,sometimes called bilateral or mir-ror symmetry, as shown at theright.
6. Does this 6-fold symmetricdesign have other lines of sym-metry? If so, draw them in.
7.Which designs in figure 1 have line symmetry? __
Draw the lines of symmetry on the designs.
8. Why do the lines of symmetry always pass through
the rotocenters? _
The editors wish to thank David Masunaga, lolani School, Honolulu, HI 96826,tor writino this iSSIJp.of thp. Nr:TM !=;t",;pnt M:>th Nntpc
Name _ Date _
ENRICHMENT 8-2ROTATIONAL SYMMETRYThe order of rotational symmetry is the number of timesthe original figure maps onto itself in a complete rotation.The figure at the right has rotational symmetry of order 5.
I!J EXERCISESWrite the order of rotational symmetry for each figure.1.
20~
4. so.80.
3.
6.
9.
10. Let a be the order of rotational symmetry and A be the angle ofrotation in degrees. Write an equation relating a and A.
Create three designs of your own. Write the order of rotational symmetry for each.11. 12. 13.
© Glencoe/McGraw-Hili 252 MathMatters 3