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Diagram showing all of the different axes of symmetry of a cube.

Axes of Symmetry of a Cube

Diagram showing all of the different axes of symmetry of a cube.

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through its center, and a similar point is found on the line at an equal distance beyond the center." — Ford, 1912

Symmetry axis

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through…

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through its center, and a similar point is found on the line at an equal distance beyond the center." — Ford, 1912

Symmetry center

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through…

A design created by inscribing 4 congruent tangent arcs in a circle.

Arcs Inscribed In A Circle

A design created by inscribing 4 congruent tangent arcs in a circle.

A design created by dividing a circle into 4 equal arcs and reflecting each arc toward the center of the circle. The arcs are inverted.

Reflected Arcs Of A Circle

A design created by dividing a circle into 4 equal arcs and reflecting each arc toward the center of…

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward the center of the circle. (The arcs are inverted.) The design is then repeated and rotated 45° to create the star-like illustration in scribed in the circle.

Reflected Arcs Of 2 Circles In A Circle

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward…

A design created by dividing a circle into 4 equal arcs and reflecting each arc toward the center of the circle. (The arcs are inverted.) The design is then repeated and rotated 45° to create the star-like illustration.

Reflected Arcs Of 2 Circles

A design created by dividing a circle into 4 equal arcs and reflecting each arc toward the center of…

A design created by dividing a circle into 4 equal arcs and reflecting each arc toward the center of the circle. (The arcs are inverted.) The design is then repeated and rotated 45° and the overlapping curves are removed to create the star-like illustration.

Reflected Arcs Of 2 Circles

A design created by dividing a circle into 4 equal arcs and reflecting each arc toward the center of…

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward the center of the circle. (The arcs are inverted.) The design is then repeated (a total of four times) and rotated 22.5° to create the star-like illustration in scribed in the circle.

Reflected Arcs Of 4 Circles In A Circle

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward…

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward the center of the circle. (The arcs are inverted.) The design is then repeated (a total of four times) and rotated 22.5° to create the star-like illustration.

Reflected Arcs Of 4 Circles In A Circle

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward…

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward the center of the circle. (The arcs are inverted.) The design is then repeated (a total of eight times) and rotated 11.25° to create the star-like illustration in scribed in the circle.

Reflected Arcs Of 8 Circles In A Circle

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward…

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward the center of the circle. (The arcs are inverted.) The design is then repeated (a total of eight times) and rotated 11.25° to create the star-like illustration.

Reflected Arcs Of 8 Circles

A design created by dividing a circle into 4 equal arcs and creating a reflection of each arc toward…

Illustration of an irregular convex decagon. This polygon has some symmetry.

Irregular Convex Decagon

Illustration of an irregular convex decagon. This polygon has some symmetry.

Illustration of an irregular convex decagon. This polygon has some symmetry.

Irregular Convex Decagon

Illustration of an irregular convex decagon. This polygon has some symmetry.

Illustration of an irregular convex decagon. This polygon has some symmetry.

Irregular Convex Decagon

Illustration of an irregular convex decagon. This polygon has some symmetry.

Illustration of an irregular convex decagon. This polygon has some symmetry.

Irregular Convex Decagon

Illustration of an irregular convex decagon. This polygon has some symmetry.

Illustration of an irregular convex decagon. This polygon has some symmetry.

Irregular Convex Decagon

Illustration of an irregular convex decagon. This polygon has some symmetry.

A 6-point star made from a non-regular concave dodecagon in which all sides are equal in length. There is vertical, horizontal, rotational, and diagonal symmetry.

Concave Equilateral Dodecagon

A 6-point star made from a non-regular concave dodecagon in which all sides are equal in length. There…

Illustration of an irregular convex dodecagon. This polygon has some symmetry.

Irregular Convex Dodecagon

Illustration of an irregular convex dodecagon. This polygon has some symmetry.

Illustration of an irregular convex dodecagon. This polygon has some symmetry.

Irregular Convex Dodecagon

Illustration of an irregular convex dodecagon. This polygon has some symmetry.

Illustration of an irregular convex dodecagon. This polygon has some symmetry.

Irregular Convex Dodecagon

Illustration of an irregular convex dodecagon. This polygon has some symmetry.

Illustration of an ellipse, whose major axis is vertical, inscribed in a circle whose diameter is equal to the length of the major axis of the ellipse. The ellipse is externally tangent to the circle.

Ellipse Inscribed In A Circle

Illustration of an ellipse, whose major axis is vertical, inscribed in a circle whose diameter is equal…

Illustration of 16 concentric congruent ellipses that are rotated about the center at equal intervals of 22.5°. The ellipses are externally tangent to the circle in which they are inscribed.

16 Rotated Concentric Ellipses

Illustration of 16 concentric congruent ellipses that are rotated about the center at equal intervals…

Illustration of 2 concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter is equal to the length of the major axes of the ellipses. The ellipses, which decrease in width in equal increments, are externally tangent to the circle. The illustration could be used as a 3-dimensional drawing of a sphere.

2 Ellipses Inscribed In A Circle

Illustration of 2 concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter…

Illustration of 2 concentric congruent ellipses that are rotated about the center at 90°. The ellipses are externally tangent to the circle in which they are inscribed.

2 Rotated Concentric Ellipses

Illustration of 2 concentric congruent ellipses that are rotated about the center at 90°. The ellipses…

Illustration of 4 concentric congruent ellipses that are rotated about the center at equal intervals of 45°. The ellipses are externally tangent to the circle in which they are inscribed.

4 Rotated Concentric Ellipses

Illustration of 4 concentric congruent ellipses that are rotated about the center at equal intervals…

Illustration of 8 concentric congruent ellipses that are rotated about the center at equal intervals of 22.5°. The ellipses are externally tangent to the circle in which they are inscribed.

8 Rotated Concentric Ellipses

Illustration of 8 concentric congruent ellipses that are rotated about the center at equal intervals…

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter is equal to the length of the major axes of the ellipses. The ellipses, which decrease in width in equal increments until the smallest one is a line, are externally tangent to the circle. The illustration could be described as a circle rotated about the poles of the vertical axis. It could also be used as a 3-dimensional drawing of a sphere.

Ellipses Inscribed In A Circle

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter…

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter is equal to the length of the major axes of the ellipses. The ellipses, which decrease in width in equal increments until the smallest one is a line, are externally tangent to the circle. The illustration could be described as a circle rotated about the poles of the vertical axis. It could also be used as a 3-dimensional drawing of a sphere.

Ellipses Inscribed In A Circle

Illustration of concentric ellipses, whose major axes are vertical, inscribed in a circle whose diameter…

In cases where the basal pinacoid is fixed by some physical property like cleavage, the elongation is in the direction of the clinodiagonal axis. This crystal of Feldspar shown is a common example of this elongation.

Elongation in the Direction of the Clinodiagonal Axis

In cases where the basal pinacoid is fixed by some physical property like cleavage, the elongation is…

Illustration of 10 congruent equilateral triangles that have the same center. Each triangle has been rotated 12° in relation to the one next to it. The outer vertices are connected with a smoother curve to form a circle. Hence, the circle is circumscribed about the triangles.

10 Congruent Rotated Equilateral Triangles

Illustration of 10 congruent equilateral triangles that have the same center. Each triangle has been…

Illustration of 10 congruent equilateral triangles that have the same center. Each triangle has been rotated 12° in relation to the one next to it.

10 Congruent Rotated Equilateral Triangles

Illustration of 10 congruent equilateral triangles that have the same center. Each triangle has been…

Illustration of 20 congruent equilateral triangles that have the same center. Each triangle has been rotated 6° in relation to the one next to it.

20 Congruent Rotated Equilateral Triangles

Illustration of 20 congruent equilateral triangles that have the same center. Each triangle has been…

Illustration of 5 congruent equilateral triangles that have the same center. Each triangle has been rotated 24° in relation to the one next to it.

5 Congruent Rotated Equilateral Triangles

Illustration of 5 congruent equilateral triangles that have the same center. Each triangle has been…

This is a diagram of the cross section of a fish, showing the bilateral symmetry of the parts: dv, dorsoventral axis; vl, right-left axis. a.p., anterior appendage; b.c., body cavity; ch, notochord; d.f., dorsal fin; g, gut; h, heart; h.a., haemal arch; m, muscles; n.., neural arch; sp, spinal cord; v.c., vertebral column.

Fish

This is a diagram of the cross section of a fish, showing the bilateral symmetry of the parts: dv, dorsoventral…

Illustration of an irregular convex heptagon. This polygon has some symmetry.

Irregular Convex Heptagon

Illustration of an irregular convex heptagon. This polygon has some symmetry.

Illustration of an irregular convex heptagon. This polygon has some symmetry.

Irregular Convex Heptagon

Illustration of an irregular convex heptagon. This polygon has some symmetry.

Illustration of an irregular convex heptagon. This polygon has some symmetry.

Irregular Convex Heptagon

Illustration of an irregular convex heptagon. This polygon has some symmetry.

Illustration of an irregular convex heptagon. This polygon has some symmetry.

Irregular Convex Heptagon

Illustration of an irregular convex heptagon. This polygon has some symmetry.

Illustration of an irregular hexagon. This is also an example of a concave polygon with symmetry.

Irregular Concave Hexagon

Illustration of an irregular hexagon. This is also an example of a concave polygon with symmetry.

Illustration of an irregular hexagon. This is also an example of a concave polygon with symmetry.

Irregular Concave Hexagon

Illustration of an irregular hexagon. This is also an example of a concave polygon with symmetry.

Illustration of an irregular hexagon. This is also an example of a concave polygon with symmetry.

Irregular Concave Hexagon

Illustration of an irregular hexagon. This is also an example of a concave polygon with symmetry.

Illustration of an irregular convex hexagon with symmetry.

Irregular Convex Hexagon

Illustration of an irregular convex hexagon with symmetry.

This illustration shows the arrangement of mirrors in a kaleidoscope (AC and BC), and the patterns formed.

Mirrors in a Kaleidoscope

This illustration shows the arrangement of mirrors in a kaleidoscope (AC and BC), and the patterns formed.

A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances from two fixed points is the square of half the distance between the points. It is a particular case of the Cassinian oval and resembles a figure 8. When the line joining the two fixed points is the axis of x and the middle point of this line is the origin, the Cartesian equation is the fourth degree equation, (((x^2)+(y^2))^2)=2(a^2)((x^2)-(y^2)). The polar equation is (ℽ^2) = 2(a^2)cos(2θ). The locus of the feet of the perpendiculars from the center of an equilateral hyperbola to its tangents is a lemniscate. The name lemniscate is sometimes given to any crunodal symmetric quartic curve having no infinite branch. The name is also sometimes given to a general class of curves derived from other curves in the way that the above is derived from the equilateral hyperbola. With these more general definitions of the lemniscate the above curve is called the lemniscate of Bernoulli.

Lemniscate

A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances…

"The symmetry of the Monoclinic System is as follows: The crystallographic axis b is an axis of binary symmetry and the plane of the a and c axes is a plane of symmetry." — Ford, 1912

Symmetry of monoclinic system

"The symmetry of the Monoclinic System is as follows: The crystallographic axis b is an axis of binary…

"The symmetry of the Normal Class of the Hexagonal System is as follows: The vertical crystallographic axis is an axis of hexagonal symmetry. There are six horizontal axes of binary symmetry, three of them being coincident with the crystallographic axes and the other three lying midway between them." — Ford, 1912

Symmetry of normal class

"The symmetry of the Normal Class of the Hexagonal System is as follows: The vertical crystallographic…

Illustration of an irregular concave octagon. This polygon has some symmetry.

Irregular Concave Octagon

Illustration of an irregular concave octagon. This polygon has some symmetry.

Illustration of an irregular concave octagon. This polygon has some symmetry.

Irregular Concave Octagon

Illustration of an irregular concave octagon. This polygon has some symmetry.

Illustration of an irregular concave octagon. This polygon has some symmetry.

Irregular Concave Octagon

Illustration of an irregular concave octagon. This polygon has some symmetry.

Illustration of an irregular convex octagon. This polygon has some symmetry.

Irregular Convex Octagon

Illustration of an irregular convex octagon. This polygon has some symmetry.

Illustration of an irregular convex octagon. This polygon has some symmetry.

Irregular Convex Octagon

Illustration of an irregular convex octagon. This polygon has some symmetry.

Illustration of an irregular convex octagon. This polygon has some symmetry.

Irregular Convex Octagon

Illustration of an irregular convex octagon. This polygon has some symmetry.

Illustration of an irregular convex octagon. This polygon has some symmetry.

Irregular Convex Octagon

Illustration of an irregular convex octagon. This polygon has some symmetry.

Illustration of an irregular pentagon. This is also an example of a concave polygon with symmetry.

Irregular Concave Pentagon

Illustration of an irregular pentagon. This is also an example of a concave polygon with symmetry.

Illustration of an irregular pentagon. This is also an example of a convex polygon with symmetry.

Irregular Convex Pentagon

Illustration of an irregular pentagon. This is also an example of a convex polygon with symmetry.

Illustration of an irregular pentagon. This is also an example of a convex polygon with symmetry.

Irregular Convex Pentagon

Illustration of an irregular pentagon. This is also an example of a convex polygon with symmetry.

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through its center, and a similar point is found on the line at an equal distance beyond the center." — Ford, 1912

Symmetry plane

"A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through…

Illustration of an irregular convex polygon with 16 sides that has symmetry. It could be used to show rotation of a square.

Irregular Convex Polygon

Illustration of an irregular convex polygon with 16 sides that has symmetry. It could be used to show…

"The symmetry of the Pyritohedral class is as follows: The three crystal axes of binary symmetry; the four diagonal axes, each of which emerges in the middle of the octant, are axes of trigonal symmetry." — Ford, 1912

Symmetry of pyritohedral class

"The symmetry of the Pyritohedral class is as follows: The three crystal axes of binary symmetry; the…