Similarity (geometry)

In Euclidean geometry, similarity describes a specific relationship between two objects. Two objects are considered similar if they possess the exact same shape. One object can be transformed into the other through a process called uniform scaling. This means you can enlarge or reduce the object by a consistent factor. You may also use translation, which is sliding the object. You can use rotation, which is turning the object. You can also use reflection, which creates a mirror image.

If these transformations allow one object to coincide precisely with another, they are similar. This concept allows mathematicians to study how shapes relate regardless of their size.

To understand the mechanism of similarity, we look at how dimensions change. A similarity transformation is a bijection that multiplies all distances by a positive real number. This number is often called the scale factor, the stretching factor, or the similarity coefficient.

When the scale factor is exactly 1, the two shapes are actually congruent. Congruent shapes are identical in both shape and size. Similarity preserves certain properties like angles, parallelism, and perpendicularity. However, it does not always preserve orientation. A direct similitude keeps the orientation the same. An opposite similitude changes the orientation through reflection.

Different types of geometric figures follow different rules for similarity. For instance, all circles are similar to one another. All squares and all equilateral triangles are also inherently similar.

However, not all rectangles or ellipses are similar. Two rectangles might have different length-to-breadth ratios. Two ellipses might have different width-to-height ratios. Even isosceles triangles can fail the similarity test if their base angles differ. For polygons with more than three sides, similarity requires more than just equal angles. For example, all rhombi are not similar even if their angles match. To guarantee similarity in polygons, the corresponding sides and diagonals must be proportional.

Triangles are unique because they have very specific similarity criteria. The AAA similarity theorem states that if two triangles have three congruent angles, they are similar. This is a mnemonic where each "A" stands for an angle. Another method is the SAS similarity criterion. This requires two pairs of sides to be proportional and the included angle to be congruent.

In Euclidean geometry, any two triangles that are similar to a third triangle are also similar to each other. This property is known as the transitivity of similarity. These rules allow for many synthetic proofs. These include the Pythagorean theorem and Ceva's theorem. Similarity also provides the essential foundation for right triangle trigonometry.

History has shaped our formal understanding of these geometric rules. John Wallis, who lived from 1616 to 1703, proposed Wallis's postulate. This postulate is logically equivalent to Euclid's parallel postulate. It relates to the existence of points that satisfy specific scaling conditions. Later, George David Birkhoff used the SAS similarity criterion in his axioms. This helped to dramatically shorten Hilbert's axioms in the study of geometry. These historical developments helped move geometry from simple observations to a rigorous axiomatic system.

Similarity also dictates how area and volume grow. The ratio between the areas of two similar figures is the square of the ratio of their corresponding lengths. If you multiply the side of a square by three, its area becomes nine times larger because three squared is nine.

The relationship for three-dimensional objects involves the cube of the ratio. If the scale factor of a solid is $k$, the volume ratio is $k^3$. For example, tripling the edge of a cube increases its volume by 27. This is known as Galileo's square-cube law. It explains how surface area and volume change at different rates as objects grow.

Finally, similarity connects to broader mathematical fields like complex arithmetic. In a 2D plane, similarity transformations can be expressed using complex numbers. This allows mathematicians to use complex arithmetic to describe rotations and scalings. Similarity also appears in various curves. Lines, circles, and parabolas all exhibit similarity. Even complex patterns like logarithmic spirals are self-similar. This means they maintain their shape as they grow or shrink, creating beautiful and infinite structures.