Editing Similarity (geometry)

{{short description|Property of objects which are scaled or mirrored versions of each other}}
{{other uses|Similarity (disambiguation)|Similarity transformation (disambiguation)}}

[[File:SimilitudeL.svg|thumb|upright=0.75|Similar figures]]
In [[Euclidean geometry]], two objects are '''similar''' if they have the same [[shape]], or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly [[scaling (geometry)|scaling]] (enlarging or reducing), possibly with additional [[translation (geometry)|translation]], [[rotation (mathematics)|rotation]] and [[reflection (mathematics)|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 [[congruence (geometry)|congruent]] to the result of a particular uniform scaling of the other.

{|cellspacing="10" 
|- valign="top"
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[[File:TranslationL.svg|thumb|upright=0.5|center|Translation]]
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[[File:RotationL.svg|thumb|upright=0.5|center|Rotation]]
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[[File:SymétrieL.svg|thumb|upright=0.5|center|Reflection]]
| 
[[File:SimilitudeHomothétieL.svg|thumb|upright=0.5|center|Scaling]]
|}

For example, all [[circle]]s are similar to each other, all [[square]]s are similar to each other, and all [[equilateral triangle]]s are similar to each other. On the other hand, [[ellipse]]s are not all similar to each other, [[rectangle]]s are not all similar to each other, and [[isosceles triangle]]s are not all similar to each other. This is because two ellipses can have different width to height ratios, two rectangles can have different length to breadth ratios, and two isosceles triangles can have different base angles. 
[[File:Similar-geometric-shapes.svg|thumb|300px|Figures shown in the same color are similar]]
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.

Two [[congruence (geometry)|congruent]] shapes are similar, with a scale factor of 1. However, some school textbooks 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.{{citation needed|date=May 2020}}

==Similar triangles==
Two triangles, {{math|△''ABC''}} and {{math|△''A'B'C'''}} are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of [[corresponding sides]] are [[proportionality (mathematics)|proportional]].{{sfn|Sibley|1998|p=35}} It can be shown that two triangles having congruent angles (''equiangular triangles'') are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem.{{sfn|Stahl|2003|p=127. This is also proved in [[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 4}} Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.<ref>For instance, {{harvnb|Venema|2006|p=122}} and {{harvnb|Henderson|Taimiņa|2005|p=123}}.</ref>

There are several criteria each of which is necessary and sufficient for two triangles to be similar:

*Any two pairs of angles are congruent,<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 4.</ref> which in Euclidean geometry implies that all three angles are congruent:{{efn|This statement is not true in [[non-Euclidean geometry]] where the triangle angle sum is not 180 degrees.}}

::If {{math|∠''BAC''}} is equal in measure to {{math|∠''B'A'C',''}} and {{math|∠''ABC''}} is equal in measure to {{math|∠''A'B'C',''}} then this implies that {{math|∠''ACB''}} is equal in measure to {{math|∠''A'C'B'''}} and the triangles are similar.

*All the corresponding sides are proportional:<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 5.</ref>

<math display=block>\frac{\overline{AB}}{\overline{A'B'}} = \frac{\overline{BC}}{\overline{B'C'}} = \frac{\overline{AC}}{\overline{A'C'}}.</math> 
:This is equivalent to saying that one triangle (or its mirror image) is an [[homothetic transformation|enlargement]] of the other.

*Any two pairs of sides are proportional, and the angles included between these sides are congruent:<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 6.</ref>
<math display=block> \frac{\overline{AB}}{\overline{A'B'}} = \frac{\overline{BC}}{\overline{B'C'}}, \quad \angle ABC \cong \angle A'B'C'.</math>
:This is known as the SAS similarity criterion.{{sfn|Venema|2006|p=143}} The "SAS" is a mnemonic: each one of the two S's refers to a "side"; the A refers to an "angle" between the two sides.

Symbolically, we write the similarity and dissimilarity of two triangles {{math|△''ABC''}} and {{math|△''A'B'C'''}} as follows:<ref name=PL>{{cite book|last1=Posamentier|first1=Alfred S.|authorlink=Alfred S. Posamentier|last2=Lehmann|first2=Ingmar|year=2012|title=[[The Secrets of Triangles]]|publisher=Prometheus Books|page=22}}</ref>

<math display=block>\begin{align}
\triangle ABC &\sim \triangle A'B'C' \\
\triangle ABC &\nsim \triangle A'B'C' 
\end{align}</math>

There are several elementary results concerning similar triangles in Euclidean geometry:{{sfn|Jacobs|1974|pp=384–393}}
* Any two [[equilateral triangle]]s are similar.
* Two triangles, both similar to a third triangle, are similar to each other ([[transitive relation|transitivity]] of similarity of triangles).
* Corresponding [[altitude (triangle)|altitudes]] of similar triangles have the same ratio as the corresponding sides.
* Two [[right triangle]]s are similar if the [[hypotenuse]] and one other side have lengths in the same ratio.<ref>{{cite book|last=Hadamard |first=Jacques|author-link=Jacques Hadamard|year=2008|title=Lessons in Geometry, Vol. I: Plane Geometry|publisher=American Mathematical Society|isbn=978-0-8218-4367-3|at=Theorem 120, p. 125|url=https://books.google.com/books?id=SaZwAAAAQBAJ&pg=PA125}}</ref> There are several equivalent conditions in this case, such as the right triangles having an acute angle of the same measure, or having the lengths of the legs (sides) being in the same proportion.

Given a triangle {{math|△''ABC''}} and a line segment {{math|{{overline|''DE''}}}} one can, with a [[straightedge and compass construction|ruler and compass]], find a point {{mvar|F}} such that {{math|△''ABC'' ~ △''DEF''}}. The statement that point {{mvar|F}} satisfying this condition exists is [[John Wallis#Geometry|Wallis's postulate]]<ref>Named for [[John Wallis]] (1616–1703)</ref> and is logically equivalent to Euclid's [[parallel postulate]].{{sfn|Venema|2006|p=122}} In [[hyperbolic geometry]] (where Wallis's postulate is false) similar triangles are congruent.

In the axiomatic treatment of Euclidean geometry given by [[George David Birkhoff]] (see [[Birkhoff's axioms]]) the SAS similarity criterion given above was used to replace both Euclid's parallel postulate and the SAS axiom which enabled the dramatic shortening of [[Hilbert's axioms]].{{sfn|Venema|2006|p=143}}

Similar triangles provide the basis for many [[synthetic geometry|synthetic]] (without the use of coordinates) proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the [[angle bisector theorem]], the [[geometric mean theorem]], [[Ceva's theorem]], [[Menelaus's theorem]] and the [[Pythagorean theorem]]. Similar triangles also provide the foundations for [[trigonometry|right triangle trigonometry]].{{sfn|Venema|2006|p=145}}

==Other similar polygons==
[[File:SimilarRectangles.svg|thumb|Similar rectangles]]
The concept of similarity extends to [[polygon]]s 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 [[proportionality (mathematics)|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 [[rhombus|rhombi]] would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all [[rectangle]]s would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.

For given {{mvar|n}}, all [[regular polygon|regular {{mvar|n}}-gons]] are similar.

==Similar curves==
Several types of curves have the property that all examples of that type are similar to each other. These include:
*[[Line (geometry)|Lines]] (any two lines are even [[congruence (geometry)|congruent]])
*[[Line segment]]s
*[[Circle]]s
*[[Parabola]]s<ref>[https://www.academia.edu/5601461/Similarity_of_Parabolas_-_A_Geometrical_Perspective a proof from academia.edu]</ref>
*[[Hyperbola]]s of a specific [[eccentricity (mathematics)|eccentricity]]<ref name="uluc">[http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node27.html The shape of an ellipse or hyperbola depends only on the ratio b/a]</ref>
*[[Ellipse]]s of a specific eccentricity<ref name="uluc" />
*[[Catenary|Catenaries]]
*Graphs of the [[logarithm]] function for different bases
*Graphs of the [[exponential function]] for different bases
*[[Logarithmic spiral]]s are self-similar

==In Euclidean space==
A '''similarity''' (also called a '''similarity transformation''' or '''similitude''') of a [[Euclidean space]] is a [[bijection]] {{mvar|f}} from the space onto itself that multiplies all distances by the same positive [[real number]] {{mvar|r}}, so that for any two points {{mvar|x}} and {{mvar|y}} we have

:<math>d(f(x),f(y)) = r\, d(x,y), </math>

where {{math|''d''(''x'',''y'')}} is the [[Euclidean distance]] from {{mvar|x}} to {{mvar|y}}.{{sfn|Smart|1998|p=92}} The [[scalar (mathematics)|scalar]] {{mvar|r}} has many names in the literature including; the ''ratio of similarity'', the ''stretching factor'' and the ''similarity coefficient''. When {{math|1=''r'' = 1}} a similarity is called an [[Euclidean plane isometry|isometry]] ([[rigid transformation]]). Two sets are called '''similar''' if one is the image of the other under a similarity.

As a map {{tmath|f : \R^n \to \R^n,}} a similarity of ratio {{mvar|r}} takes the form

:<math>f(x) = rAx + t,</math>

where {{tmath|A \in O^n(\R)}} is an {{math|''n'' × ''n''}} [[orthogonal matrix]] and {{tmath|t \in \R^n}} is a translation vector.

Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments.{{sfn|Yale|1968|p=47 Theorem 2.1}} Similarities preserve angles but do not necessarily preserve orientation, ''direct similitudes'' preserve orientation and ''opposite similitudes'' change it.{{sfn|Pedoe|1988|pp=179–181}}

The similarities of Euclidean space form a [[group (mathematics)|group]] under the operation of composition called the ''similarities group'' {{mvar|S}}.{{sfn|Yale|1968|p=46}} The direct similitudes form a [[normal subgroup]] of {{mvar|S}} and the [[Euclidean group]] {{math|''E''(''n'')}} of isometries also forms a normal subgroup.{{sfn|Pedoe|1988|p=182}} The similarities group {{mvar|S}} is itself a subgroup of the [[affine group]], so every similarity is an [[affine transformation]].
<!-- FOR LATER INCLUSION
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. -->

One can view the Euclidean plane as the [[complex plane]],{{efn|This traditional term, as explained in its article, is a misnomer. This is actually the 1-dimensional complex line.}} that is, as a 2-dimensional space over the [[real number|reals]]. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by 
*<math>f(z) = az + b</math> (direct similitudes), and 
*<math>f(z) = a\overline z + b</math> (opposite similitudes), 
where {{mvar|a}} and {{mvar|b}} are complex numbers, {{math|''a'' ≠ 0}}. When {{math|1={{abs|''a''}}= 1}}, these similarities are isometries.

== Area ratio and volume ratio ==
[[File:A proportion to conceive square root of 5.svg|left|thumb|upright=1.26
|{{anchor|enlargedRightTriangle}}The [[tessellation]] of the large triangle shows that it is similar to the small triangle with an area ratio of 5. The similarity ratio is <math>\tfrac{5}{h} = \tfrac{h}{1} = \sqrt 5.</math> This can be used to construct a [[pinwheel tiling|non-periodic infinite tiling]].]]

{{Main|Square–cube law}}
The ratio between the [[area]]s 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 three, its area is multiplied by nine — i.e. by three squared). The altitudes of similar triangles are in the same ratio as corresponding sides. If a triangle has a side of length {{mvar|b}} and an altitude drawn to that side of length {{mvar|h}} then a similar triangle with corresponding side of length {{mvar|kb}} will have an altitude drawn to that side of length {{mvar|kh}}. The area of the first triangle is <math>A = \tfrac{1}{2}bh,</math> while the area of the similar triangle will be 
<math display=block>A' =\frac{1}{2} \cdot kb \cdot kh = k^2A.</math>
Similar figures which can be decomposed into similar triangles will have areas related in the same way. The relationship holds for figures that are not rectifiable as well.

The ratio between the [[volume]]s 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 three, its volume is multiplied by 27 — i.e. by three cubed).

Galileo's square–cube law concerns similar solids. If the ratio of similitude (ratio of corresponding sides) between the solids is {{mvar|k}}, then the ratio of surface areas of the solids will be {{math|''k''<sup>2</sup>}}, while the ratio of volumes will be {{math|''k''<sup>3</sup>}}.

{{Clear}}

==Similarity with a center==
{{multiple image
 | align = left | total_width = 720
 | image_gap = 15
 | image1 = 3 iterations of similarities applied to 3 regular polygons.svg
 | image2 = Academ Example of similarity with ratio square root of 2.svg
 | caption1 = Example where each similarity<br />[[function composition|composed]] with itself several times successively<br />has a '''center''' at the center of a [[regular polygon]] that it shrinks.
 | caption2 = {{anchor|caption2OfSection}}Example of direct similarity of center {{mvar|S}}<br />[[function composition|decomposed]] into a rotation of 135° angle<br />and a homothety that halves [[area]]s.
 | footer = <span style="font-size:120%">Examples of direct similarities that have each a '''center'''.</span>
 | footer_align = center
}}
{{Clear}}
If a similarity has exactly one [[invariant (mathematics)|invariant point]]: a point that the similarity keeps unchanged, then this only point is called "'''center'''" of the similarity.

On the first image below the title, on the left, one or another similarity shrinks a [[regular polygon]] into a [[concentric objects|concentric one]], the vertices of which are each on a side of the previous polygon. This rotational reduction [[iterated function|is repeated]], so the initial polygon is extended into an [[mise en abyme|abyss]] of regular polygons. The '''center''' of the similarity is the common center of the successive polygons. A red [[line segment|segment]] joins a vertex of the initial polygon to its [[Image (mathematics)#Image of an element|image]] under the similarity, followed by a red segment going to the following image of vertex, and so on to form a [[spiral]]. Actually we can see more than three direct similarities on this first image, because every regular polygon is invariant under certain direct similarities, more precisely certain rotations the center of which is the center of the polygon, and a composition of direct similarities is also a direct similarity. For example we see the image of the initial regular [[pentagon]] under a [[homothety]] of negative ratio {{mvar|−k}}, which is a similarity of ±180° angle and a positive ratio equal to {{mvar|k}}.

Below the title on the right, the second image shows a similarity [[function composition|decomposed]] into a [[rotation (mathematics)|rotation]] and a homothety. Similarity and rotation have the same angle of +135 degrees [[modular arithmetic|modulo 360 degrees]]. Similarity and homothety have the same ratio of {{tmath|\tfrac{\sqrt 2}{2},}} [[multiplicative inverse]] of the ratio {{tmath|\sqrt 2}} ([[square root of 2]]) of the [[inverse function|inverse]] similarity. Point {{mvar|S}} is the common '''center''' of the three transformations: rotation, homothety and similarity. For example point {{mvar|W}} is the image of {{mvar|F}} under the rotation, and point {{mvar|T}} is the image of {{mvar|W}} under the homothety, more briefly
<math display=block>T = H(W) = (R(F)) = (H \circ R)(F) = D(F),</math>
by naming {{mvar|R}}, {{mvar|H}} and {{mvar|D}} the previous rotation, homothety and similarity, with “{{mvar|D}}" like "Direct".

This direct similarity that transforms triangle {{math|△''EFA''}} into triangle {{math|△''ATB''}} can be decomposed into a rotation and a homothety of same center {{mvar|S}} in several manners. For example, {{math|1=''D'' = ''R'' ○ ''H'' = ''H'' ○ ''R''}}, the last decomposition being only represented on the image. To get {{mvar|D}} we can also compose in any order a rotation of −45° angle and a homothety of ratio {{tmath|\tfrac{- \sqrt 2}{2}.}}

With "{{mvar|M}}" like "Mirror" and "{{mvar|I}}" like "Indirect", if {{mvar|M}} is the [[reflection (mathematics)|reflection]] with respect to line {{mvar|CW}}, then {{math|1=''M'' ○ ''D'' = ''I''}} is the '''indirect''' similarity that transforms segment {{mvar|{{overline|BF}}}} like {{mvar|D}} into segment {{mvar|{{overline|CT}}}}, but transforms point  {{mvar|E}} into {{mvar|B}} and point  {{mvar|A}} into  {{mvar|A}} itself. Square {{mvar|ACBT}} is the image of {{mvar|ABEF}} under similarity {{mvar|I}} of ratio {{tmath|\tfrac{1}{\sqrt 2}.}} Point  {{mvar|A}} is the center of this similarity because any point {{mvar|K}} being invariant under it fulfills <math>AK = \tfrac{AK}{\sqrt 2},</math> only possible if {{math|1=''AK'' = 0}}, otherwise written {{math|1=''A'' = ''K''}}.

How to [[#caption2OfSection|construct the center {{mvar|S}}]] of direct similarity {{mvar|D}} from square {{mvar|ABEF}}, how to find point {{mvar|S}} center of a rotation of +135° angle that transforms ray {{tmath|\overset{}\overrightarrow{SE} }} into ray {{tmath|\overset{}\overrightarrow{SA} }}? This is an [[inscribed angle]] problem plus a question of [[orientation (vector space)|orientation]]. The set of points {{mvar|P}} such that <math>\overset{}{ \overrightarrow{PE}, \overrightarrow{PA} = +135^\circ }</math> is an arc of circle {{mvar|{{overarc|EA}}}} that joins  {{mvar|E}} and  {{mvar|A}}, of which the two radius leading to  {{mvar|E}} and  {{mvar|A}} form a [[central angle]] of {{math|1=2(180° − 135°) = 2 × 45° = 90°}}.  This set of points is the blue quarter of circle of center {{mvar|F}} inside square {{mvar|ABEF}}. In the same manner, point {{mvar|S}} [[set (mathematics)|is a member]] of the blue quarter of circle of center {{mvar|T}} inside square {{mvar|BCAT}}. So point {{mvar|S}} is the [[intersection (Euclidean geometry)|intersection]] point of these two quarters of circles.

==In general metric spaces==
[[File:Sierpinski deep.svg|thumb|300px|[[Sierpiński triangle]]. A space having self-similarity dimension <math>\tfrac{\log 3}{\log 2} = \log_2 3,</math> which is approximately 1.58. (From [[Hausdorff dimension]].)]]

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

<math display=block>d(f(x),f(y)) = r d(x,y).</math>

Weaker versions of similarity would for instance have {{mvar|f}} be a bi-[[Lipschitz continuity|Lipschitz]] function and the scalar {{mvar|r}} a limit

<math display=block>\lim \frac{d(f(x),f(y))}{d(x,y)} = r. </math>

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 {{math|(''X'', ''d'')}} is a set {{mvar|K}} for which there exists a finite set of similitudes {{math|{ ''f{{sub|s}}''}{{sub|''s''∈''S''}}}} with contraction factors {{math|0 ≤ ''r{{sub|s}}'' < 1}} such that {{mvar|K}} is the unique compact subset of {{mvar|X}} for which

:[[File:Epi17.png|thumb|A self-similar set constructed with two similitudes: 
<math>\begin{align} 
z' &= 0.1[(4+i)z+4] \\
z' &= 0.1[(4+7i)z^* + 5 - 2i]
\end{align}</math>]]

<math display=block>\bigcup_{s\in S} f_s(K)=K.</math>

These self-similar sets have a self-similar [[measure (mathematics)|measure]] {{math|''μ{{sup|D}}''}} with dimension {{mvar|D}} given by the formula

<math display=block>\sum_{s\in S} (r_s)^D=1 </math>

which is often (but not always) equal to the set's [[Hausdorff dimension]] and [[packing dimension]]. If the overlaps between the {{math|''f{{sub|s}}''(''K'')}} are "small", we have the following simple formula for the measure:

<math display=block>\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.\,</math>

==Topology==
{{more citations needed section|date=August 2018}}
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
# Positive defined:
<math display=block>\forall (a,b), S(a,b)\geq 0</math>
# Majored by the similarity of one element on itself ('''auto-similarity'''):
<math display=block>S (a,b) \leq S (a,a) \quad \text{and} \quad \forall (a,b), S (a,b) = S (a,a) \Leftrightarrow a=b</math>

More properties can be invoked, such as:
* '''Reflectivity''': <math>\forall (a,b)\ S (a,b) = S (b,a),</math> or
* '''Finiteness''': <math>\forall (a,b)\ S(a,b) < \infty.</math> 
The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).

Note that, in the topological sense used here, a similarity is a kind of [[measure (mathematics)|measure]]. This usage is '''not''' the same as the ''similarity transformation'' of the {{section link||In Euclidean space}} and {{section link||In general metric spaces}} sections of this article.

==Self-similarity==
[[Self-similarity]] means that a pattern is '''non-trivially similar''' to itself, e.g., the set {{math|{..., 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12, ...} }} of numbers of the form {{math|{2{{sup|''i''}}, 3·2{{sup|''i''}}} }} where {{mvar|i}} ranges over all integers. When this set is plotted on a [[logarithmic scale]] it has one-dimensional [[translational symmetry]]: adding or subtracting the logarithm of two to the logarithm of one of these numbers produces the logarithm of another of these numbers. In the given set of numbers themselves, this corresponds to a similarity transformation in which the numbers are multiplied or divided by two.

==Psychology==
{{Expand section|date=July 2021}}
The intuition for the notion of geometric similarity already appears in human children, as can be seen in their drawings.<ref>{{cite thesis|last=Cox|first=Dana Christine|year=2008|type=Ph.D.|url=https://scholarworks.wmich.edu/cgi/viewcontent.cgi?article=1762&context=dissertations&httpsredir=1&referer=|title=Understanding Similarity: Bridging Geometric and Numeric Contexts for Proportional Reasoning|location=Kalamazoo, Michigan |publisher=Western Michigan University|isbn=978-0-549-75657-6|s2cid=61331653}}</ref> Certain perceptual categorization models in psychology are based on geometric similarity, assuming that learning involves the storage of specific instances (''i.e.'' of general object specifications) in memory. The categorization of another object ist subsequently based on the similarity of the object to the instances in memory.<ref>{{Cite book |title=The Psychology of Learning and Motivation: Advances in Research and Theory |date=2004 |publisher=Elsevier textbooks |isbn=978-0-08-052277-7 |editor-last=Ross |editor-first=Brian H. |edition=1. Aufl |location=s.l. |page=231}}</ref>

==See also==
* [[Basic proportionality theorem]]
* [[Congruence (geometry)]]
* [[Hamming distance]] (string or sequence similarity)
* [[Helmert transformation]]
* [[Homoeoid]] (shell of concentric, similar ellipsoids)
* [[Inversive geometry#Dilations|Inversive geometry]]
* [[Jaccard index]]
* [[Nearest neighbor search|Similarity search]]
* [[Proportionality (mathematics)|Proportionality]]
* [[Semantic similarity]]
* [[Similarity (philosophy)#Transformation|Similarity (philosophy)]]
* [[Similarity space]] on [[numerical taxonomy]]
* [[Solution of triangles]]
* [[Spiral similarity]]

==Notes==
{{reflist}}
{{reflist|group=lower-alpha}}

==References==
* {{cite book|last1=Henderson|first1=David W.|author-link1=David W. Henderson|last2=Taimiņa|first2=Daina|author-link2=Daina Taimiņa|year=2005|title=Experiencing Geometry/Euclidean and Non-Euclidean with History|edition=3rd|publisher=Pearson Prentice-Hall|isbn=978-0-13-143748-7}}
* {{cite book|last=Jacobs|first=Harold R.|author-link=Harold R. Jacobs|year=1974|title=Geometry|publisher=W. H. Freeman and Co.|isbn=0-7167-0456-0}}
* {{cite book|last=Pedoe|first=Dan|author-link=Daniel Pedoe|year=1988|orig-year=1970|title=Geometry/A Comprehensive Course|publisher=Dover|isbn=0-486-65812-0}}
* {{cite book|last=Sibley|first=Thomas Q.|year=1998|title=The Geometric Viewpoint/A Survey of Geometries|publisher=Addison-Wesley|isbn=978-0-201-87450-1|url-access=registration|url=https://archive.org/details/geometricviewpoi0000sibl}}
* {{cite book|last=Smart|first=James R.|year=1998|title=Modern Geometries|edition=5th|publisher=Brooks/Cole|isbn=0-534-35188-3}}
* {{cite book|last=Stahl|first=Saul|year=2003|title=Geometry/From Euclid to Knots|publisher=Prentice-Hall|isbn=978-0-13-032927-1}}
* {{cite book|last=Venema|first=Gerard A.|year=2006|title=Foundations of Geometry|publisher=Pearson Prentice-Hall|isbn=978-0-13-143700-5}}
* {{cite book|last=Yale|first=Paul B.|year=1968|title=Geometry and Symmetry|publisher=Holden-Day}}

==Further reading==
* {{cite book|last=Cederberg|first=Judith N.|year=2001|orig-year=1989|title=A Course in Modern Geometries|chapter=Chapter 3.12: Similarity Transformations|publisher=Springer|pages=183–189|isbn=0-387-98972-2}}
* [[Harold Scott MacDonald Coxeter|Coxeter, H. S. M.]] (1969) [1961]. "§5 Similarity in the Euclidean Plane". pp.&nbsp;67–76. "§7 Isometry and Similarity in Euclidean Space". pp.&nbsp;96–104. ''Introduction to Geometry''. [[John Wiley & Sons]].
* {{cite book|last=Ewald|first=Günter|year=1971|title=Geometry: An Introduction|publisher=[[Wadsworth Publishing]]|pages=106, 181}}
* {{cite book|last=Martin|first=George E.|year=1982|chapter=Chapter 13: Similarities in the Plane|title=Transformation Geometry: An Introduction to Symmetry|publisher=Springer|pages=136–146|isbn=0-387-90636-3}}

==External links==
{{commons category|Similarity (geometry)}}
*[http://www.mathopenref.com/similartriangles.html Animated demonstration of similar triangles]
*[http://dynamicmathematicslearning.com/fundamental-theorem-similarity.html A Fundamental Theorem of Similarity] - an illustrative dynamic geometry sketch

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[[Category:Geometry]]
[[Category:Equivalence (mathematics)]]
[[Category:Euclidean geometry]]
[[Category:Triangle geometry]]