Mirror symmetry

The name of speculative symmetry comes from the image obtained by reflecting light on a flat surface. There are many examples of symmetry to speculate both in nature and in artificial objects.

In geometry, mirror symmetry (also known as bilateral or reflection symmetry), is a transformation with respect to a plane of symmetry, in which each point of a figure is associated with another point called image, which meets the following conditions:

a) The distance of a point and its image to the plane of symmetry is the same.

b) The segment that joins a point with its image is perpendicular to the plane of symmetry.

A figure that remains invariant when subjected to reflection is said to possess mirror or reflection symmetry. In the case of figures on a two-dimensional plane, the plane of symmetry becomes an axis of symmetry.

String theory

The mirror symmetry is a relationship that can exist between two varieties of Calabi-Yau. It can usually be between two such six-dimensional manifolds, whose shapes may appear quite different geometrically, but which are nevertheless equivalent when used as hidden dimensions of string theory. More specifically, mirror symmetry relates two varieties M and W whose Hodge numbers

h^1.1. and h^1.2

are swapped. String theory compacted into these two manifolds can be shown to lead to identical physical phenomena.

The discovery of mirror symmetry is tied to names such as Brian Greene, Ronen Plesser, Philip Candelas, Monika Lynker, Rolf Schimmrigk and others. Andrew Strominger, Shing-Tung Yau, and Eric Zaslow have shown that mirror symmetry is a special example of duality T: the Calabi-Yau manifold can be described as a bundle whose fiber is a three-dimensional torus. The simultaneous action of duality T in the three dimensions of this torus is equivalent to mirror symmetry.

Mirror symmetry allows many calculations to be simplified by invoking the mirror image of a given physical situation, which can often be much easier to solve.

Analytic Geometry

• The point (x,y,z) and (-x,y,z) are symmetrical regarding the Oyz plane
• The point (x,y,z) and (x,-y,z) are symmetrical regarding the Oxz plane
• The point (x,y,z) and (x,y, -z) are symmetrical regarding the Oxy plane

Geometry of space

• The cube is a symmetrical figure regarding the plane that passes through the straights that contain the diagonals of two opposite faces.

• The cube is a symmetrical figure regarding the plane passing through the midpoints of the four perpendicular edges to two opposite sides.

• The plane that contains two diagonals of the cube is a plane of symmetry of the cube.

Examples

Figures with the axes of symmetry drawn. The figure without axis is asymmetric

In two dimensions, symmetry is verified with respect to a straight line or axis of symmetry, and in three dimensions with respect to a plane of symmetry. An object or figure that is indistinguishable from its transformed image is called a mirror image. Briefly, a line of symmetry divides the shape into two exact halves that can be folded together.

Symmetric functions

Graphic of a normal distribution. Known as the bell of Gauss, it is an example of symmetrical function

In formal terms, a mathematical object is symmetric with respect to an operation defined as reflection, rotation, or translation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a property given of the object form a group. Two objects are symmetric to each other with respect to a given set of operations if one is obtained from the other by some of the operations (and vice versa).

The two-dimensional graph of a function is said to be symmetric if there exists at least one straight line or axis such that all its perpendiculars that intersect the graph at a certain distance from the axis also intersect it at opposite direction at the same distance.

Another way to think of a symmetric function is that if the shape were folded in half about the axis, the two halves would be identical: the two halves are mirror images of each other.

Therefore, a square has four lines of symmetry, because there are four different ways to fold it and make all the edges match up. A circle has infinitely many lines of symmetry.

Symmetrical geometric shapes

2D forms with speculative symmetry

Trapecio isosceles y deltoide
Let's go.
Eighteen

Triangles with reflection symmetry are isosceles triangles. Quadrilaterals with reflection symmetry are deltoids, deltoids (concave), rhombi, and isosceles trapezoids. All even-sided regular polygons have two simple reflection shapes, one with reflection axes through opposite vertices and one with axes through the center of opposite sides.

For an arbitrary shape, the axiality of the shape measures its proximity to being bilaterally symmetric. It is equal to 1 for shapes with reflection symmetry and between 2/3 and 1 for any convex shape.

Mathematical Equivalents

For each axis or plane of reflection, the symmetry group is isomorphic with C_s (see point groups in three dimensions), one of the three types of order two (involutions), therefore algebraically C₂. The fundamental domain is a half-plane or half-space.

In certain contexts, there is a symmetry of rotation and reflection. So mirror image symmetry is equivalent to inversion symmetry; in such contexts in modern physics, the term parity or P-symmetry is used for both.

Advanced types of reflection symmetry

For more general types of reflection, there are consequently more general types of reflection symmetry. For example:

• With respect to a non-isometric aphin revolution (an oblique symmetry with respect to a line, plane, etc.)
• With respect to a circular investment.

In nature

Many animals, like this majoid Maja crispata, are bilaterally symmetrical

Bilaterally symmetrical animals possess reflection symmetry in the sagittal plane, which divides the body vertically into left and right halves, with one of each pair of sensory organs and limbs on each side. Most animals are bilaterally symmetrical, probably because this supports forward movement and simplifies their structure.

In architecture

Speculative symmetry is often used in architecture, as in the facade of the Basilica of Santa Maria Novella, Florence (1470)

Mirror symmetry is often used in architecture, such as on the façade of the Basilica of Santa Maria Novella, Florence. It is also found in the design of ancient structures such as Stonehenge. Symmetry was a central element in some styles of architecture, such as Palladianism.