The modern word to which similarity belongs is an evaluation of the consistency of elements such as fine structure or nature between two individuals or parts.It is widely used in mathematics, structure, model, chemistry, language and other fields.
Chinese name
Similarity
Foreign name
similarity
application area
Mathematics, physics, graphics, structure, chemistry, language
In Euclidean geometry, if two objects have the same shape, or if the shape of one object is the same as the mirror image of the other, the two shapes are similar.In fact, if any shape can be completely coincident with another shape after translation, scaling, rotation and flipping, the two shapes are similar.
Two triangles with equal triangles and proportional three sides are called similar triangles
Similar triangle is one of the important proof models in geometry, which is the generalization of congruent triangle.A congruent triangle can be understood as a similar triangle with a similarity ratio of 1.Similar triangle is actually a set of theorems, which mainly describes the relationship between sides and angles of two triangles in geometry.
Properties of similar triangles:
Define that the corresponding angles of similar triangles are equal and the corresponding sides are proportional.
Theorem The ratio of any corresponding line segment of a similar triangle is equal to the similarity ratio.
Theorem The area ratio of similar triangles is equal to the square of the similarity ratio.
Determination of similar triangles:
By analogy with the judgement theorem of congruent triangles, the following conclusions can be drawn:
Theorem Two triangles with equal angles are similar.
Theorem Two triangles whose sides are proportional and whose angles are equal are similar.
Theorem Two triangles with three sides in proportion are similar.
Theorem Two right triangles whose right side is proportional to the hypotenuse are similar.
Special cases of similar triangles:
1. All congruent triangles are similar
A congruent triangle is a special similar triangle with a similarity ratio of 1.On the contrary, when the similarity ratio is 1, the similar triangle is congruent.
2. Two isosceles triangles with equal top or bottom angles are similar.Thus, all equilateral triangles are similar.
Two similar polygons are called similar polygons.When the corresponding sides of two polygons are proportional and the corresponding angles are equal, they are similar.Two convex polygons with equal number of sides must be similar.The perimeter ratio of two similar polygons is equal to their similarity ratio, and the area ratio is equal to the square of the similarity ratio.
In topology, metric spaces can be constructed by defining similarity rather than distance.Similarity is a function whose value is greater when two points are closer.
(4) Self similarity
•Self similarityIt means that the pattern is very similar to itself, for example, {2i，3•2i}The formal number set {..., 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12,...}, where i covers all integers.When this set is plotted on a logarithmic scale, it has one-dimensional translational symmetry: adding or subtracting the logarithms of two numbers and the logarithms of one of them will produce the logarithms of the other number.In a given set of numbers themselves, this corresponds to a similarity transformation in which numbers are multiplied or divided by 2.
•fractal, has the morphological characteristics of filling space in the form of non integer dimensions.It is usually defined as "a rough or fragmentary geometric shape, which can be divided into several parts, and each part is (at least approximately) a reduced shape of the whole", that is, it has the property of self similarity.The word Fractal was coined by Mandelbrot, and its original meaning is irregular and fragmented.In 1973, B.B. Mandelbrot first proposed the idea of fractal dimension and fractal dimension when he gave a lecture in the French Academy.
Similarity of matrices
In linear algebra, if there is an irreversible n-by-n matrix P, it is called similarmatrixA and B
B=P-1AP
Transition A ↦ P−1AP is called the similarity transformation or conjugate of matrix A.Therefore, in general linear groups, similarity is equal to conjugation, and similarity matrix is also called conjugate matrix;However, in a given subgroup H of a general linear group, the concept of conjugate may be more restrictive than similar concepts, because it requires P to be located in H[2]。
Similarity in Language
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Similarity of words
In linguistics, lexical similarity is a measure of the similarity of word sets in two given languages.The lexical similarity of 1 (or 100%) means that words overlap completely, while 0 means that there are no similar words.Different methods are used to define lexical similarity, and the results are correspondingly different.
WordsSyntagmatic similarity can be used to evaluate the degree of genetic relationship between two languages.A percentage higher than 85% usually indicates that the two languages being compared are likely to be related dialects.Lexical similarity is only an expression of mutual understanding between two languages, because the latter also depends on the degree of phonetic, morphological and grammatical similarity.Changes due to different word lists have an impact on this.For example, French and English have considerable similarities in the field of culture related vocabulary, while in terms of basic (function) words, their similarities are relatively small.Unlike mutual understanding, lexical similarity can only be symmetrical[1]。
Semantic similarity
Semantic similarity is a measure defined on a group of documents or terms, in which the differences between items are based on the similarity of their meaning or semantic content, rather than the similarity of dictionaries.These are mathematical tools used to estimate the strength of semantic relationships between language units, concepts or instances, and numerical descriptions obtained by comparing information that supports their meaning or describes their nature.The word semantic similarity is often confused with semantic relevance.Semantic relevance includes any relationship between two words, while semantic similarity only includes "yes a" relationship.For example, "car" is similar to "bus", but also related to "road" and "driving".
In terms of computation, semantic similarity can be estimated by defining topological similarity, and the distance between terms/concepts can be defined by using ontology.For example, a naive metric used to compare concepts sorted in a partial order set and represented as nodes of a directed acyclic graph (for example, a taxonomy) would be the shortest path connecting two concept nodes.On the basis of text analysis, statistical methods such as vector space model can also be used to estimate the semantic relevance between language units (such as words and sentences), so as to associate words and text context from an appropriate text corpus.The evaluation of the proposed semantic similarity/relevance measure is mainly carried out in two ways.The former is based on a dataset designed by experts and consists of word pairs with semantic similarity/relevance estimation.The second method is based on the integration of measures in specific applications such as information retrieval, recommendation system, natural language processing, etc.
Similarity in Chemistry
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Chemical similarity (or molecular similarity) refers to the structural similarity of two elements, molecules or compounds, or the similarity of their effects when participating in chemical reactions.If the biological effect and its similarity are discussed, the biological activity of the compound is generally used. Otherwise, the activity of the compound will be used to measure the effect of participating in the chemical reaction.
The concept of chemical similarity (or molecular similarity) is one of the most important topics in chemical informatics.Chemical similarity plays an important role in the modern research of predicting the properties of compounds or designing compounds with specific properties.Some drug design studies will use large chemical databases for screening, which is also related to chemical similarity.The above research is based on Johnson and Maggiora's law of similar properties: "similar compounds will have similar properties".
Other Similarities
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Structural similarity
Structural similarity index (SSIIndex) is an index used to measure the similarity of two digital images. When one of the two images is undistorted and the other is distorted, the structural similarity of the two images can be regarded as a measure of the image quality of distorted images[3]。
Genetic similarity
The academic community often uses theDNA sequence[4]Share the percentage of human sequence to indicate similarity.This figure shows the percentage of base pairs that are the same between the two species.Here is a list of genetic similarities relative to humans and data sources.
These data come from different secondary data sources and are obtained by different methods (such as DNA DNA hybridization or sequence alignment), which may lead to different results for comparison between the same species.Therefore, these data should only be used for approximate similarity.
