What is a lateral symmetry?

Answer 1

Something that is symmetrical along a line. Like you could fold it in half and it would be the same. Like a rorschach inkblot. A butterfly. A person.

The other kind of symmetry is radial symmetry, which is symmetrical from a point. Like a starfish. I think things that are radially symmetrical are always also laterally symmetrical, but there could be some weird exception to that that I can't think of.

Answer 2

Lateral symmetry isn't a standard mathematical term, but it is widely enough used that it can be understood to have a specific meaning.

Lateral symmetry is simply symmetry (i.e., [geometrical] "balance", here meaning "congruity", but sometimes meaning "similarity") that is lateral (i.e., of or related to the side).

A rectangle and an isosceles triangle both have bilateral symmetry (i.e, you can draw a line down the center and both sides are the same). An equilateral triangle has both bilateral and trilateral symmetry.

(Bilateral symmetry isn't the same as line symmetry, though it's often used as a synonym; both point symmetry and line symmetry include bilateral symmetry, though in Biology, bilateral symmetry is used to designate only certain cases of line symmetry, and none of point symmetry. More on this later, but a non-rhombus parallelogram has bilateral symmetry through any straight line that splits it in half, without having any line symmetry at all.)

In geometry there are several types of symmetry:

Line (aka reflectional) symmetry is symmetry through a line. All the shapes mentioned above have line symmetry.

Think of a piece of paper with a red side and a green side. Imagine you cut out a shape from the paper and trace the shape on another piece of paper when the red side of the shape is visible. If you can turn over the paper so that the green shape is visible and still place it perfectly in the tracing, the shape has line symmetry.

Now imagine you draw a perfect circle around the tracing with at least two points of the tracing on the circumference of the circle and no points of the tracing outside of the circle. Imagine you place the shape perfectly in its tracing, and push a pin the center of the circle, so that you can perfectly rotate the shape in the circle. If you can rotate the shape by less than 360 degrees and at the end of the rotation have it perfectly inside the tracing, the shape has rotational symmetry.

(Radial symmetry is a biological term, not a geometric one. In biology, symmetry refers to somewhat closely approximated geometrical symmetry. Radial symmetry refers to somewhat closely approximated rotational symmetry. A starfish has radial symmetry. A seahorse doesn't, because even though it slightly resembles an S with perfect rotational symmetry, there's significant difference in the details of it's top and bottom.)

If you can rotate the shape by exactly 180 degrees and still have it perfectly inside the circle, the shape has point symmetry. (I.e. Point symmetry is a specific case of rotational symmetry.)

Lateral symmetry can be line symmetry, point symmetry, rotational symmetry, or any combination. However, in biology, the terms radial symmetry and bilateral symmetry are used to represent mutually exclusive cases, even though, e.g., starfish, which are classified as being radially symmetrical are have 5 instances of line symmetry as well in the geometrical sense. Sometimes the geometrical line symmetry of a starfish is even more closely approximated than it's closely approximated rotational symmetry, but all starfish are classified as having radial symmetry.)

In general conversation as well, the word "symmetry" need not refer to precise geometric symmetry. A tourist might say that a bridge spanning from north to south is symmetric even if the west fence is a foot taller than the east fence. In contrast, an engineer designing the same bridge before the addition of the (relatively light) side fences might have been careful to ensure that east and west sides of the bridge itself, being constructed of identical material, were as close as possible to sharing precise geometrical symmetry.

Other geometrical types of symmetry are translational symmetry, and scale symmetry. Neither of these are directly related to lateral symmetry.

All of the types of symmetry mentioned so far are basic and two dimensional cases. They can be expanded into three or even more dimensions. For example, two dimensional line symmetry can be expanded into three dimensional plane symmetry. Point symmetry can be expanded to three dimensional line symmetry.

(Three dimensional plane symmetry is what you appear to share with your reflection in a flat mirror.Three dimensional line symmetry is what you appear to share with your reflection in the corner of two mirrors at a right angle.)

Lateral symmetry in three dimensions would therefore be another type of symmetry, such as that possessed by any of the 5 platonic solids.

In addition, any two or more basic types of symmetry can be combined to form yet another type of symmetry. For example, concentric symmetry is a specific case combination of scale and translation symmetry.

Answer 3

Lateral symmetry isn't a standard mathematical term, but it is widely enough used that it can be understood to have a specific meaning.

Lateral symmetry is simply symmetry (i.e., [geometrical] "balance", here meaning "congruity", but sometimes meaning "similarity") that is lateral (i.e., of or related to the side).

A rectangle and an isosceles triangle both have bilateral symmetry (i.e, you can draw a line down the center and both sides are the same). An equilateral triangle has both bilateral and trilateral symmetry.

(Bilateral symmetry isn't the same as line symmetry, though it's often used as a synonym; both point symmetry and line symmetry include bilateral symmetry, though in biology, bilateral symmetry is used to designate only certain cases of line symmetry, and none of point symmetry. More on this later, but a non-rhombus parallelogram has bilateral symmetry through any straight line that splits it in half, without having any line symmetry at all.)

In geometry there are several types of symmetry:

Line (aka reflectional) symmetry is symmetry through a line. All the shapes mentioned above, with the exception of the non-rhombus parallelogram, have line symmetry.

Think of a piece of paper with a red side and a green side. Imagine you cut out a shape from the paper and trace the shape on another piece of paper when the red side of the shape is visible. If you can turn over the paper so that the green shape is visible and still place it perfectly in the tracing, the shape has line symmetry.

Now imagine you draw a perfect circle around the tracing with at least two points of the tracing on the circumference of the circle and no points of the tracing outside of the circle. Imagine you place the shape perfectly in its tracing, and push a pin the center of the circle, so that you can perfectly rotate the shape in the circle. If you can rotate the shape by less than 360 degrees and at the end of the rotation have it perfectly inside the tracing, the shape has rotational symmetry.

(Radial symmetry is a biological term, not a geometric one. In biology, symmetry refers to somewhat closely approximated geometrical symmetry. Radial symmetry refers to somewhat closely approximated rotational symmetry. A starfish has radial symmetry. A seahorse doesn't, because even though it slightly resembles an S with perfect rotational symmetry, there's significant difference in the details of it's top and bottom.)

If you can rotate the shape by exactly 180 degrees and still have it perfectly inside the circle, the shape has point symmetry. (I.e. Point symmetry is a specific case of rotational symmetry.)

Lateral symmetry can be line symmetry, point symmetry, rotational symmetry, or any combination. However, in biology, the terms radial symmetry and bilateral symmetry are used to represent mutually exclusive cases, even though, e.g., starfish, which are classified as being radially symmetrical are have 5 instances of line symmetry as well in the geometrical sense. Sometimes the geometrical line symmetry of a starfish is even more closely approximated than it's closely approximated rotational symmetry, but all starfish are classified as having radial symmetry.)

In general conversation as well, the word "symmetry" need not refer to precise geometric symmetry. A tourist might say that a bridge spanning from north to south is symmetric even if the west fence is a foot taller than the east fence. In contrast, an engineer designing the same bridge before the addition of the (relatively light) side fences might have been careful to ensure that east and west sides of the bridge itself, being constructed of identical material, were as close as possible to sharing precise geometrical symmetry.

Other geometrical types of symmetry are translational symmetry, and scale symmetry. Neither of these are directly related to lateral symmetry.

All of the types of symmetry mentioned so far are basic and two dimensional cases. They can be expanded into three or even more dimensions. For example, two dimensional line symmetry can be expanded into three dimensional plane symmetry. Point symmetry can be expanded to three dimensional line symmetry.

(Three dimensional plane symmetry is what you appear to share with your reflection in a flat mirror.Three dimensional line symmetry is what you appear to share with your reflection in the corner of two mirrors at a right angle.)

Lateral symmetry in three dimensions would therefore be another type of symmetry, such as that possessed by any of the 5 platonic solids.

In addition, any two or more basic types of symmetry can be combined to form yet another type of symmetry. For example, concentric symmetry is a specific case combination of scale and translation symmetry.