تشابه (رياضيات)

Similar figures

نقول عن شكلان أنهما متشابهان إذا كان أحدهما مطابق للآخر بعد إجراء تحجيم عليه (تكبير أو تصغير)، مع دوران أو نقل إضافيين للحصول على الاتجاه الصحيح المطابق للشكل الأصلي. مثال: جميع الدوائر هي أشكال متشابهة لبعضها البعض لأنها تختلف فقط في نصف القطر، كما أن جميع المربعات متشابهة لبعضلها البعض، ولكن ليس جميع القطوع الناقصة مشابهة لبعضها البعض، كذلك الأمر بالنسبة للقطوع الزائدة.

Translation

Rotation

Reflection

Scaling

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, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. This is because two ellipses can have different width to height ratio, two rectangle can also have a different length to breadth ratio, and two isosceles triangle can have different base angles.

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 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.^{[بحاجة لمصدر]}

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Area ratio and volume ratio

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

5/ h = h /1 = \sqrt 5

. This can be used to construct a non-periodic infinite tiling.

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 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 $b$ and an altitude drawn to that side of length $h$ then a similar triangle with corresponding side of length $kb$ will have an altitude drawn to that side of length $kh$ . The area of the first triangle is, $A = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/2bh$ , while the area of the similar triangle will be $A' = 1 / 2 (kb)(kh) = k 2 A$ . 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 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 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 $k$ , then the ratio of surface areas of the solids will be $k 2$ , while the ratio of volumes will be $k 3$ .

Similarity with a center

Example where each similarity
composed with itself several times successively
has a center at the center of a regular polygon that it shrinks.

Example of direct similarity of center S
decomposed into a rotation of 135° angle
and a homothety that halves areas.

Examples of direct similarities that have each a center.

If a similarity has exactly one invariant point: a point that the similarity keeps unchanged, then this only point is called "center" of the similarity.

أنواع التشابه

التشابه في المستوي المركب : ( نوعان تشابه مباشر وغير مباشر )

التشابه المباشر :

هو كل تحويل نقطي معرّف بـ :

Z'= a × z + b

حيث : العدد a ينتمي إلى مجموعة الأعداد المركبة C، وطويلة a تساوي 1 : |a| = 1

هو تشابه مباشر مركزه W لاحقة b/1-a و زاويته عمدة a، arg(a) ونسبته k=a

يرمز له بـ : $S(W,Q,k)$

التشابه غير المباشر : هو أحد أنواع التشابه وهو من التحويلات النقطية الرياضية.

التشابه في المثلثات

يكون مثلثان متشابهان في المستوي إذا وفقط إذا كانت قياس زوايهما الثلاثة متساوية، ولكن وعلى اعتبار أن مجموع قياس الزوايا الثلاثة في المثلث ثابت ويساوي 180 درجة فيكفي أن تكون زاويتين متساويتين ليكون المثلثان متشابهان.

الطبولوجيا

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:
$\forall (a,b),S(a,b)\geq 0$
Majored by the similarity of one element on itself (auto-similarity):
$S(a,b)\leq S(a,a)\quad {\text{and}}\quad \forall (a,b),S(a,b)=S(a,a)\Leftrightarrow a=b$

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).

Note that, in the topological sense used here, a similarity is a kind of measure. This usage is not the same as the similarity transformation of the § In Euclidean space and § 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 ${\dots, 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12, \dots}$ of numbers of the form ${2 i, 3\cdot2 i}$ where $i$ 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

The intuition for the notion of geometric similarity already appears in human children, as can be seen in their drawings.^[1]

Notes

^ Cox, Dana Christine (2008). Understanding Similarity: Bridging Geometric and Numeric Contexts for Proportional Reasoning (Ph.D.). Kalamazoo, Michigan: Western Michigan University. ISBN 978-0-549-75657-6. S2CID 61331653.

الكلمات الدالة:

رياضيات

[1] Cox, Dana Christine (2008). Understanding Similarity: Bridging Geometric and Numeric Contexts for Proportional Reasoning (Ph.D.). Kalamazoo, Michigan: Western Michigan University. ISBN 978-0-549-75657-6. S2CID 61331653.

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