Similarity (geometry)

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Figures shown in the same color are similar

Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a uniform scaling of the other.

For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other.

If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.

This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.

Similar triangles[edit]

Two triangles \triangle ABC and \triangle DEF are said to be similar if any of the following equivalent conditions hold:

1. They have two identical angles, which implies that their angles are all identical. For instance:

 \angle BAC is equal in measure to \angle EDF, and \angle ABC is equal in measure to \angle DEF. This also implies that \angle ACB is equal in measure to \angle DFE.

2. Corresponding sides have lengths in the same ratio:

 {AB \over DE} = {BC \over EF} = {AC \over DF}. This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.

3. Two sides have lengths in the same ratio, and the angles included between these sides have the same measure. For instance:

 {AB \over DE} = {BC \over EF} and \angle ABC is equal in measure to \angle DEF.

When two triangles \triangle ABC and \triangle DEF are similar, one writes

\triangle ABC\sim\triangle DEF \,

or

\triangle ABC \, ||| \,\triangle DEF \,

Other similar polygons[edit]

The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles (otherwise, for example, all rhombi would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all rectangles would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.

Similar curves[edit]

Several types of curves have the property that all examples of that type are similar to each other. These include:

Similarity in Euclidean space[edit]

One of the meanings of the terms similarity and similarity transformation (also called dilation) of a Euclidean space is a function f from the space into itself that multiplies all distances by the same positive scalar r, so that for any two points x and y we have

d(f(x),f(y)) = r d(x,y), \,

where "d(x,y)" is the Euclidean distance from x to y. Two sets are called similar if one is the image of the other under such a similarity.

A special case is a homothetic transformation or central similarity: it neither involves rotation nor taking the mirror image. A similarity is a composition of a homothety and an orthogonal transformation. Therefore, in general Euclidean spaces every similarity is an affine transformation, because the Euclidean group E(n) is a subgroup of the affine group.

Viewing the complex plane as a 2-dimensional space over the reals, the 2D similarity transformations expressed in terms of complex arithmetic are f(z)=az+b and f(z)=a\overline z+b where a and b are complex numbers, a ≠ 0.

Galileo's square-cube law[edit]

Main article: Square-cube law

The ratio between the areas of similar figures is equal to the square of the ratio of corresponding lengths of those figures (for example, when the side of a square or the radius of a circle is multiplied by two, its area is multiplied by four — i.e. by two squared).

The ratio between the volumes of similar figures is equal to the cube of the ratio of corresponding lengths of those figures (for example, when the edge of a cube or the radius of a sphere is multiplied by two, its volume is multiplied by eight — i.e. by two cubed).

Similarity in general metric spaces[edit]

Sierpinski triangle. A space having self-similarity dimension ln 3 / ln 2 = log23, which is approximately 1.58. (from Hausdorff dimension.)

In a general metric space (Xd), an exact similitude is a function f from the metric space X into itself that multiplies all distances by the same positive scalar r, called f's contraction factor, so that for any two points x and y we have

d(f(x),f(y)) = r d(x,y).\, \,

Weaker versions of similarity would for instance have f be a bi-Lipschitz function and the scalar r a limit

\lim \frac{d(f(x),f(y))}{d(x,y)} = r.

This weaker version applies when the metric is an effective resistance on a topologically self-similar set.

A self-similar subset of a metric space (Xd) is a set K for which there exists a finite set of similitudes \{ f_s \}_{s\in S} with contraction factors 0\leq r_s < 1 such that K is the unique compact subset of X for which

\bigcup_{s\in S} f_s(K)=K. \,

These self-similar sets have a self-similar measure \mu^Dwith dimension D given by the formula

\sum_{s\in S} (r_s)^D=1 \,

which is often (but not always) equal to the set's Hausdorff dimension and packing dimension. If the overlaps between the f_s(K) are "small", we have the following simple formula for the measure:

\mu^D(f_{s_1}\circ f_{s_2} \circ \cdots \circ f_{s_n}(K))=(r_{s_1}\cdot r_{s_2}\cdots r_{s_n})^D.\,

Topology[edit]

In topology, a metric space can be constructed by defining a similarity instead of a distance. The similarity is a function such that its value is greater when two points are closer (contrary to the distance, which is a measure of dissimilarity: the closer the points, the lesser the distance).

The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are

  1. Positive defined: \forall (a,b), S(a,b)\geq 0
  2. Majored by the similarity of one element on itself (auto-similarity): S (a,b) \leq S (a,a) and \forall (a,b), S (a,b) = S (a,a) \Leftrightarrow a=b

More properties can be invoked, such as reflectivity (\forall (a,b)\ S (a,b) = S (b,a)) or finiteness (\forall (a,b)\ S(a,b) < \infty). The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).

Self-similarity[edit]

Self-similarity means that a pattern is non-trivially similar to itself, e.g., the set {.., 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12, ..}. When this set is plotted on a logarithmic scale it has translational symmetry.

See also[edit]

References[edit]

  • Judith N. Cederberg (1989, 2001) A Course in Modern Geometries, Chapter 3.12 Similarity Transformations, pp. 183–9, Springer ISBN 0-387-98972-2 .
  • H.S.M. Coxeter (1961,9) Introduction to Geometry, §5 Similarity in the Euclidean Plane, pp. 67–76, §7 Isometry and Similarity in Euclidean Space, pp 96–104, John Wiley & Sons.
  • Günter Ewald (1971) Geometry: An Introduction, pp 106, 181, Wadsworth Publishing.
  • George E. Martin (1982) Transformation Geometry: An Introduction to Symmetry, Chapter 13 Similarities in the Plane, pp. 136–46, Springer ISBN 0-387-90636-3 .

External links[edit]