# Talk:Similarity (geometry)

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Field: Geometry

## Matrix congruency

"Two real matrices A and B are called congruent if there is a regular real matrix P such that ${\displaystyle P^{t}AP=B}$. Two real symmetric matrices are similar to each other if and only if they are congruent"

This is clearly false: take the 1-by-1 matrices A=1 and P=2, then A cannot be similar to B=4 since they fail to have the same determinant.

I removed the second sentence, as that seems to be the wrong one.--Patrick 10:58, 2 October 2005 (UTC)

## Graphic request

I just added a graphic to this article. If you need something else, please be explicit. John Reid 22:24, 14 April 2006 (UTC)

## Matrix similarity

This subject needs its own article and we need disambiguation page. Same with other subjects which just happen to have the same name, but nothing to do with the concept of similarity in Euclidean geometry. --345Kai 21:27, 19 April 2006 (UTC)

After the article split, I'm wondering where two of the "See also" links should go, namely Hamming distance and Jaccard index. Both of them are similarity measures of vectors. I'm not sure they fit well in there given current content of that article, but they are definitely measures of similarity. Perhaps they might fit best in some third yet unwritten similarity article for vectors and strings (strings meaning from a computer science perspective)? Halcyonhazard 03:16, 25 September 2006 (UTC)

## Similarity between parabolas

How is it that all parabolas are similar? After all, ${\displaystyle y=x^{2}}$ and ${\displaystyle y=4x^{2}}$ differ by a scaling in one coordinate direction, not by a scaling in all directions equally. Ted 20:20, 13 March 2007 (UTC)

Consider the part of the graph of
${\displaystyle y=x^{2}\,}$
that is between x = −1 and x = 1. You could say that changing x to 2x squeezes this horizontally to
${\displaystyle y=(2x)^{2}=4x^{2}\,}$
between x = −1/2 and x = 1/2, and that is NOT similar to the original graph, i.e. it is not similar to the part of the parabola between x = 1 and x = −1. However, the part of the new graph that is between x = −1/4 and x = 1/4 is similar to the part of the old graph between x = −1 and x = 1. Consider any point (ab) on the graph of y = 4x2. Let c = 4a and d = 4b. Then d = 4b = 4(4a2) = 16a2 = (4a)2 = c2. So if the point (ab) is on the graph of y = 4x2, then the point (4a, 4b) is on the graph of y = x2. The transformation
${\displaystyle (a,b)\mapsto (4a,4b)\,}$
scales in all directions equally. Michael Hardy 22:17, 13 March 2007 (UTC)
Here's another way to see it. To rescale y = ax2 to coincide with y = bx2, transform y → (a/b)y and x → (a/b)x. Then [(a/b)y] = b[(a/b)x]2. Duoduoduo (talk) 19:27, 27 July 2011 (UTC)
To clarify that, we want to transform ${\displaystyle y=ax^{2}}$ into ${\displaystyle y^{\prime }=bx^{\prime 2}.}$ Let ${\displaystyle y^{\prime }=(a/b)y}$ and ${\displaystyle x^{\prime }=(a/b)x.}$ Then ${\displaystyle y^{\prime }=(a/b)y=(a/b)ax^{2}=(a/b)a\cdot [(b/a)x^{\prime }]^{2}=bx^{\prime 2}.}$ Duoduoduo (talk) 14:01, 11 November 2013 (UTC)

Excuse me for possibly being dense, but how does the lead image help illustrate the concept of similarity? Some of the figures with the same colors are similar; others are not. If there is a point here, a caption that points it out is urgently needed. –Henning Makholm 22:59, 12 June 2007 (UTC)

Hm, on further observation, perhaps the centres of the dark contour lines are in fact similar for same-coloured figures. The contour widths do not scale together with the figure, which made e.g. the two orange ellipses appear non-similar to me at first. A caption would still be an improvement. –Henning Makholm 23:06, 12 June 2007 (UTC)

## Congruence and Similarity

The article reads, "A concept commonly taught in high school mathematics is that of proving the 'angle' and 'side' theorems, which can be used to define two triangles as similar (or indeed, congruent)." Are congruent shapes similar? On the one hand, this sentence and the various similarity theorems seem to assume so. Dilations defined as a function f such that d(f(x),f(y)) = r d(x,y) show that size-preserving transformations are just dilations with r = 1. On the other hand, both this article and the article on scaling say that similarity and scaling involve enlarging or shrinking, which implies that congruent shapes have not been scaled and are not similar. Some school mathematics textbooks specifically exclude the possibility of congruence in their definitions and examples of scaling. Which is it? Is their consensus? 128.62.149.81 (talk) 14:30, 16 October 2008 (UTC)

If you want similarity to be an equivalence relation, then you have to accept that congruency is a type of similarity. From a perspective of a theory I think this is a strong and compelling reason to accept congruency as a type of similarity. I am not sure why a theory would need to have congruent triangles not be similar.
On this occasion, I agree with anon editor 72.178.193.150. I have always assumed (and taught) that congruence is a special type of similarity. British school text books allow a scale factor of unity. Is this not the case in the USA? Dbfirs 01:55, 30 July 2009 (UTC)
I agree that it makes more sense to allow congruent shapes to be similar. But is this what mathematicians actually say? I am glad to hear that they say so in British school text books. French school text books (those I have seen) quite specifically deny that congruent shapes are similar, requiring that a dilation for r different than 1 be applied between similar shapes. USA textbooks I have examined seem vague and even contradictory. No American school textbook I have seen directly addresses the question.--seberle (talk) 21:35, 4 August 2009 (UTC)
It all depends on which definition of similarity is used. I was amazed to find that the BBC website defines similar shapes as specifically different in size. Nevertheless, most academic sites include the possibility of congruence in their definitions of similarity. Dbfirs 22:30, 4 August 2009 (UTC)
I would have to take exception with the statement, "No American school textbook I have seen directly addresses the question". While it may be true for that editor, I have checked four Geometry books (1959, 1963, 1991 and 1996) and they each address the question of interest here.
Schaum's Outlines: Geometry (1963, revised 3rd Ed. 2000) - Similar polygons are polygons whose corresponding angles are congruent and whose corresponding sides are in proportion. Similar polygons have the same shape although not necessarily the same size.
Geometry for Enjoyment and Challenge (1991) - Similar polygons are polygons in which 1) The ratios of the measures of corresponding sides are equal 2) Corresponding angles are congruent.
So far, I have not encountered any contradictory definitions. JackOL31 (talk) 02:50, 18 October 2009 (UTC)
You are probably right. My initial exposure was to geometry lessons for younger American children, and some are confused. I don't have the contradictory book with me, but take this definition from the Math Is Fun! site as a typical vague example: "In Geometry, two shapes are Similar if the only difference is size." But most secondary school textbooks seem to be consistent. --seberle (talk) 03:22, 18 October 2009 (UTC)
You can probably find an elementary text with contradictory information. Many of those books are written by elementary educators who are not mathematicians, per se. Also, while simplifying concepts for younger minds one may inadvertantly introduce inaccuracies. I find the example you gave acceptable for young minds. With the picture given on the website, I think a young student would get the basic concept. It would be too difficult to explain ratio of similitude and so forth to them without the risk of having them become glassy-eyed. Realistically, they're not going to remember the definition, anyway. Well, at least not beyond the upcoming test! JackOL31 (talk) 22:04, 18 October 2009 (UTC)
I agree to some extent. But it is possible to be intuitive and still be accurate. Another web site I saw yesterday said congruent polygons are similar, but similar polygons are not congruent. The kids may get the right idea, but it is still erroneous. French textbooks for elementary children I have seen are intuitive and precise at the same time: Congruent shapes are not similar; the ratio must be different than one. Why do the French believe 9-year-olds are capable of understanding these fine distinctions, but not Americans? Well, anyway, I agree they're not going to remember the definition anyway. Of more importance are the examples and activities the children do. The French include activities helping young children to learn that congruence is excluded from the concept of similarity (by their definition). What do Americans do, other than plant seeds for future misunderstandings? --seberle (talk) 23:44, 18 October 2009 (UTC)
Whoa, I think there is a misunderstanding here. I thought you were referring to the previous definition that should have read, "Congruent polygons are similar, but similar polygons are not necessarily congruent." I agree with the definitions I posted above illustrating that congruency is a subset of similarity. In the definition on this webpage, there is no requirement that r is different from 1. The only requirement is that everything is in proportion, and a 1:1 ratio is valid. From this seat, it seems that the French plant seeds for future misunderstandings. JackOL31 (talk) 21:46, 21 October 2009 (UTC)
No, the French do not define similarity the same way. At least not in the math textbooks I have seen. They are quite explicit about excluding congruence. The contradiction between the Anglo-Saxon definition and the French definition was the source of my initial confusion. --seberle (talk) 00:41, 22 October 2009 (UTC)
I understand what you are saying, but I have difficulties following that approach. First, the equation presented on this page holds for r=1. Any restriction on positive r seems to me to be mathematically arbitrary. Secondly, the closely related Similar-Figures Theorem states that for similar figures, the ratio of their areas equals the square of the ratio of their corresponding segment lengths. Assuming we had two similar figures with one pair of corresponding sides having lengths of 12 and 9 units, then the ratio of the figure's areas would be (12/9)^2 = (4/3)^2 = 16/9 or 16:9. Again, this theorem places no restriction on the ratio of similitude and works quite well for a ratio of 1/1. Next, in an effort to reduce nationality bias, I searched for a non-US reference on similarity and found the following book online, "The Ontario High School Geometry - Theoretical, Authorized by the Minister of Education for Ontario, Toronto - The Copp, Clark Company, Limited 1910 First Ed.". This book gives the following definition: Book V, 131 Definition. If two polygons of the same number of sides have the angles of one taken in order around the figure respectively equal to the angles of the other in order, and have also the corresponding sides in proportion, the polygons are said to be similar polygons. Again, there is no restriction. Lastly, it just seems to me that if something is not similar, then it is dissimilar. Following the approach you suggest, if one has an equilateral triangle with a perimeter of 6 units, then one would be required to place another equilateral triangle with a perimeter of 6 units in with circles, squares, quadrilaterals and other dissimilar figures. That would be a difficult concept for me to accept. JackOL31 (talk) 21:51, 24 October 2009 (UTC)
Well, I was a bit surprised too. Please note that it is not I suggesting this approach. I am just reporting on how this concept is being taught in French elementary schools. As noted above, certain Wikipedia articles implied the same for the English-speaking world (by saying the shape must be enlarged or shrunk), so I was asking for clarification. I agree that that the English approach makes more sense, but I don't think we have much say in the matter, do we? In a similar vein, I think the common English definition of trapezoid/trapezium which excludes parallelograms creates the same unnecessary complications as the French definition of similarity. Another example would be the traditional definition of an isosceles triangle which excludes equilateral triangles. All of these definitions which exclude a special case cause complications which are unnecessary in my mind. But in the end, I have no control over how these terms are defined. --seberle (talk) 22:30, 24 October 2009 (UTC)
Perhaps it is a matter of interpretation. When I think of enlarged or shrunk, I don't eliminate the congruent size as we go from one extreme of similarity to the other. On a different note, I must say I am surprised by your views regarding trapezoids and isosceles triangles. I also believe that the set of set of quadrilaterals having 2 pairs of parallel sides is a subset of quadrilaterals having 1 pair of parallel sides (no claim on the adjacent sides). I generally avoid this discussion because of the intensity that the "discussion" can generate. If I can't convince someone that an equilateral triangle is just midway point of all isosceles triangles for a given isosceles side length, then I have no hope of convincing anyone regarding parallelograms as trapezoids. It might interest you to know that www.mathopenref.com has definitions that agree with our perceptions. JackOL31 (talk) 00:54, 3 November 2009 (UTC)
I don't think you can ever convince someone of a mathematical definition because a definition is inherently arbitrary -- determined by tradition, perhaps an established authority, or just maintained as "my" definition. A mathematical object is whatever mathematicians define it to be. But you can certainly argue for the value of a certain definition or the superiority of a new definition. --seberle (talk) 02:56, 3 November 2009 (UTC)
I'm fully in agreement with JackOL31 on this (and also on the trapezium controversy), and I was amazed that the French, and a BBC website, disagree. One would expect that we mathematicians, of all people, could agree on our definitions, but it seems that some of us (myself included on triangles) stick with tradition rather than inclusion. I suppose that, following Wikipedia policy, we have to report both views. Dbfirs 09:18, 3 November 2009 (UTC)
Because this is the English-language version of Wikipedia, I think a limitation to just English definitions is sufficient. It is already confusing enough describing American/British differences such as trapezoid/trapezium without pulling in idiosyncrasies in other languages. --seberle (talk) 15:48, 3 November 2009 (UTC)
Let me make it clear that I am in agreement with everyone here. I am strongly in favor of inclusive definitions. But sometimes tradition, or a textbook I am working with, force me to use an exclusive definition. What I am saying is that these things are usually out of my control. --seberle (talk) 15:57, 3 November 2009 (UTC)
Does that prevent you from presenting the alternative as a point of discussion? It could lead to some interesting thoughts on the matter. Regarding similar/dissimilar, perhaps congruency is asimilar as in neither similar nor dissimilar. I wonder how many in a class would side with dissimilar, how many with similar, and how many with asimilar. JackOL31 (talk) 23:13, 3 November 2009 (UTC)
If I am in control of the class, I certainly can present another view and I agree it would lead to interesting conversations. I was the one who started this thread (I didn't realize I wasn't logged in). The event that triggered my questions was this: I am French but went to American schools. I had a daughter in CM1 (American 4th grade) last year. In conversation about the math curriculum with her teacher, I learned the curriculum clearly taught that r must never be equal to 1 for dilations, which surprised me. I found some other curricula more vague, including possible contradictions on different Wikipedia sites, so I wrote for clarification. The teacher understands my position, but clearly I am not in a position to change the French curriculum. In other situations, certainly I could at least dialog. --seberle (talk) 03:08, 4 November 2009 (UTC)
Re: Dbfirs. I think there is general agreement on the core concepts. It's just the various nuances that spark disagreements. I don't know if it's important for everyone to have the same view on every little thing. What is important is to be open to the possible change of position. JackOL31 (talk) 23:13, 3 November 2009 (UTC)
Re:Seberle - the few times that I ran into disagreements with my children's teacher, I just asked my children follow what they were told for class purposes, but provided the correct explanation to my child at home. I learned this the hard way when I wrote a note on my son's paper asking why the "empty set" was the wrong answer. The reply was that the correct answer was "no answer possible". JackOL31 (talk) 01:33, 15 November 2009 (UTC)
I was always careful to phrase questions in such a way as to avoid these controversies, and to allow a variety of correct responses to open-ended questions, but it seems that not all teachers follow this practice. "No answer possible" sounds like a poor selection of multiple-choice responses? Dbfirs 08:04, 15 November 2009 (UTC)

## Complex map

Currently the article contains the phase:

and all affine transformations are of the form ${\displaystyle f(z)=az+b{\overline {z}}+c}$ (a, b, and c complex).

This statement seems faulty: summing two similar images may not be similar. Suppoose a = b, then the figure collapses to a line.Rgdboer (talk) 23:18, 29 August 2012 (UTC)

The phrase was removed today.Rgdboer (talk) 22:41, 7 September 2012 (UTC)

## Similar triangles

I have again removed,

4. Both triangles are right triangles, and the hypotenuse and one other side have lengths in the same ratio. This case differs from 3. above because a right angle can never be the included angle between the hypotenuse and one side.

from this section for two reasons. First of all, it is unsourced and secondly it doesn't belong in this listing as it is not a general property of similar triangles the way the other three items are. I have spent a considerable amount of time and effort into finding sources for all of the material in this section and would like to maintain that level of citation. If a source can be found for the above, I would happily see it put back in the article in the paragraph which deals with specific properties a little further down. Notice that I am not claiming that the statement is not true, it is a simple consequence of the Pythagorean theorem and the SAS similarity criterion, its truth is not in doubt. It needs a citation to show that it is anything more than a homework exercise. Bill Cherowitzo (talk) 05:24, 24 July 2014 (UTC)

Your comment is valid but your action is not. On Wikipedia, when we see a statement that needs an in-line citation we place the {{Citation needed}} tag adjacent to the statement. See Wikipedia:Citation needed where it provides general guidance. It gives only one grounds for immediate deletion of an unsourced statement: Controversial, poorly-sourced claims in biographies of living people. Clearly, the article on similarity is not a biography about a living person, and nor is it controversial.
After a statement has been tagged "Citation needed" for a suitable length of time, it may be erased but there is nothing written anywhere on Wikipedia to support the idea that immediate deletion of an unsourced statement is appropriate except in the case of biographies of living people.
Please revert your deletion of the statement regarding RHS. Insert the Citation needed tag if you must. Then I am willing to discuss the subject.
Alternatively, you could ask for a third opinion by raising the matter at WP:3O. Dolphin (t) 07:55, 24 July 2014 (UTC)

Ok. I put it back where I think it belongs, but I would still like to see a citation. Bill Cherowitzo (talk) 16:21, 24 July 2014 (UTC)

I think the section on similar triangles should be organized to differentiate between:

• Properties of similar triangles, and
• Criteria which are sufficient to prove that two triangles are similar.

The sentence There are several statements which imply that two triangles are similar is inappropriate language for an encyclopedia article on a math topic. I suggest this would be more encyclopedic if it said: “There are numerous criteria that are sufficient to prove that two triangles are similar:”

Any two equilateral triangles are similar. This is a property of equilateral triangles but it isn’t a property of similar triangles. Nevertheless, I have no objection to it being included with the other properties of similar triangles.

Transitivity of similar triangles is a property of similar triangles.

Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. This is a property of similar triangles. However, I would prefer “Similar triangles have areas in the same ratio as the square of the ratio of corresponding sides.”

Two right triangles are similar if the hypotenuse and one other side have lengths in the same ratio. RHS is one of the criteria that is sufficient to prove similarity of two right triangles. It is not a property of similar triangles, and not even a property of similar right triangles. It is not the same criteria as the SAS Similarity Criterion because SAS requires knowledge of the included angle. In the Wikipedia article on congruence of geometric figures, RHS is included as one of four criteria that are sufficient to prove two triangles are congruent – see Congruence (geometry)#Determining congruence.

The section presently identifies four criteria that are sufficient to prove similarity:

AA
SSS
SAS
RHS

The section presently displays confusion over the distinction between properties and criteria for sufficiency. It would be good if this appearance of confusion could be eliminated. Dolphin (t) 11:30, 26 July 2014 (UTC)