Congruence (geometry) in the context of "Face-transitive"

A regular polyhedron is a polyhedron with regular and congruent polygons as faces. Its symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: a tetrahedron (four faces), a cube (six faces), an octahedron (eight faces), a dodecahedron (twelve faces), and an icosahedron (twenty faces).

Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the Timaeus, that the classical elements were made of these regular solids.

In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon (or a sphere or ellipsoid inscribed in a convex polyhedron) is tangent to every side or face of the outer figure (but see Inscribed sphere for semantic variants). A polygon inscribed in a circle, ellipse, or polygon (or a polyhedron inscribed in a sphere, ellipsoid, or polyhedron) has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure. An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed figure gives another inscribed figure that is congruent to the original one.

Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, and triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a cyclic polygon, and the circle is said to be its circumscribed circle or circumcircle.

The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment and comes from the ancient Greek ἀπόθεμα ("put away, put aside"), made of ἀπό ("off, away") and θέμα ("that which is laid down"), indicating a generic line written down. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent.

In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions (isotropically). The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.

More general is scaling with a separate scale factor for each axis direction. Non-uniform scaling (anisotropic scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an oblique angle, or when the shadow of a flat object falls on a surface that is not parallel to it.

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form ⁠${\displaystyle \textstyle ax^{2}+bx+c}$⁠ to the form ⁠${\displaystyle \textstyle a(x-h)^{2}+k}$⁠ for some values of ⁠${\displaystyle h}$⁠ and ⁠${\displaystyle k}$⁠. In terms of a new quantity ⁠${\displaystyle x-h}$⁠, this expression is a quadratic polynomial with no linear term. By subsequently isolating ⁠${\displaystyle \textstyle (x-h)^{2}}$⁠ and taking the square root, a quadratic problem can be reduced to a linear problem.

The name completing the square comes from a geometrical picture in which ⁠${\displaystyle x}$⁠ represents an unknown length. Then the quantity ⁠${\displaystyle \textstyle x^{2}}$⁠ represents the area of a square of side ⁠${\displaystyle x}$⁠ and the quantity ⁠${\displaystyle {\tfrac {b}{a}}x}$⁠ represents the area of a pair of congruent rectangles with sides ⁠${\displaystyle x}$⁠ and ⁠${\displaystyle {\tfrac {b}{2a}}}$⁠. To this square and pair of rectangles, one more square is added, of side length ⁠${\displaystyle {\tfrac {b}{2a}}}$⁠. This crucial step completes a larger square of side length ⁠${\displaystyle x+{\tfrac {b}{2a}}}$⁠.

In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion.

Motions can be divided into direct (also known as proper or rigid) and indirect (or improper) motions.Direct motions include translations and rotations, which preserve the orientation of a chiral shape.Indirect motions include reflections, glide reflections, and Improper rotations, that invert the orientation of a chiral shape.Some geometers define motion in such a way that only direct motions are motions.