Dilations and symmetry
Dilations change a figure's size but not its shape, producing similar figures. Symmetry describes the transformations that leave a figure looking unchanged.
Dilations
A dilation with centre O and scale factor k moves each point along the ray from O so that its distance from O is multiplied by k. With centre at the origin, \((x, y) \to (kx, ky)\): a scale factor of 3 maps (2, -1) to (6, -3). If k > 1 the image is an enlargement; if 0 < k < 1, a reduction.
Properties of dilations
A dilation preserves angles and the shape of a figure but multiplies every length by k, so the image is similar to the pre-image. Lines not through the centre map to parallel lines. Areas are multiplied by k squared.
Similarity transformations
Two figures are similar exactly when one can be mapped onto the other by a dilation combined with rigid transformations. This gives a precise meaning to 'same shape, different size', just as rigid transformations define congruence.
Line symmetry
A figure has line (reflectional) symmetry if a reflection over some line maps it onto itself. An isosceles triangle has one line of symmetry, a rectangle two, a square four, and a regular n-gon n lines. A circle has infinitely many.
Rotational symmetry
A figure has rotational symmetry if a rotation of less than 360 degrees maps it onto itself. A regular pentagon has rotational symmetry of order 5, mapping onto itself every 72 degrees. A parallelogram has rotational symmetry of order 2 (180 degrees) but, unless it is a rectangle or rhombus, no line symmetry.
Symmetry in the world
Snowflakes typically show sixfold symmetry because of the hexagonal arrangement of water molecules in ice. Many flowers show fivefold symmetry. Logos, tilings, and architecture use symmetry for balance, and crystallographers classify crystals by their symmetries.
Tessellations
A tessellation covers a plane with shapes and no gaps or overlaps. Translations, rotations, and reflections generate repeating patterns; mathematicians proved there are exactly 17 distinct types of repeating symmetry patterns in the plane, called wallpaper groups. The artist M. C. Escher used them to create interlocking designs of birds, fish, and lizards.
Worked example
A photo of triangle ABC with A(0, 0), B(4, 0), and C(0, 3) is enlarged with centre at the origin and scale factor 2.5. The image has vertices (0, 0), (10, 0), and (0, 7.5). The original hypotenuse is 5, so the image's is 12.5; the original area of 6 becomes 6 x 2.5 squared = 37.5.
Key takeaways
A dilation with scale factor k multiplies distances from its centre by k; centred at the origin it maps (x, y) to (kx, ky), enlarging when k > 1 and reducing when 0 < k < 1. Dilations preserve angles, so images are similar, with lengths times k and areas times k squared. Similar figures are related by dilations plus rigid transformations. Line symmetry maps a figure onto itself by reflection, and rotational symmetry by a rotation under 360 degrees; a regular n-gon has n of each. Tessellations repeat shapes with no gaps, in 17 wallpaper groups, as Escher's art shows.
Flashcard
Scales distances from a centre by k
Check your understanding
A dilation centred at the origin with scale factor 3 maps (2, -1) to: