symmetry

symmetry

Let

V

be a

Euclidean vector space

,

F

⊆

V

, and

E

:

V

→

V

be a

Euclidean transformation

that is not the

identity map

.

The following terms are used to indicate that

E

⁢

(

F

)

=

F

if

E

is a

rotation

:

•

F

has

rotational symmetry

;

•

F

has

point symmetry

;

•

F

has

symmetry about a point

;

•

F

is

symmetric about a point

.

If

V

=

ℝ

2

, then the last two terms may be used to indicate the specific case in which

E

is

conjugate

to

(

-

1

0

0

-

1

)

, i.e. (

http://planetmath.org/Ie

) the angle of rotation is

180

∘

.

The following are classic examples of rotational symmetry in

ℝ

2

:

•

Regular polygons

: A

regular

n

-gon is symmetric about its center (

http://planetmath.org/Center9

) with valid angles of rotation

θ

=

(

360

⁢

k

n

)

∘

for any positive integer

k

<

n

.

•

Circles: A circle is symmetric about its center (

http://planetmath.org/Center8

) with uncountably many valid angles of rotation.

As another example, let

F

=

⋃

k

=

1

4

P

k

, where each

P

k

is defined thus:

P

1

=

{

(

x

,

y

)

:

0

≤

x

≤

4

1

+

3

⁢

and

⁢

(

2

-

3

)

⁢

x

≤

y

≤

x

}

,

P

2

=

{

(

x

,

y

)

:

4

1

+

3

≤

x

≤

2

⁢

and

⁢

x

≤

y

≤

(

2

+

3

)

⁢

x

-

4

}

,

P

3

=

{

(

x

,

y

)

:

2

≤

x

≤

4

⁢

3

1

+

3

⁢

and

⁢

(

-

2

+

3

)

⁢

x

+

8

-

4

⁢

3

≤

y

≤

(

-

2

-

3

)

⁢

x

+

4

+

4

⁢

3

}

,

P

4

=

{

(

x

,

y

)

:

4

⁢

3

1

+

3

≤

x

≤

4

⁢

and

⁢

(

-

2

+

3

)

⁢

x

+

8

-

4

⁢

3

≤

y

≤

-

x

+

4

}

.

Then

F

has point symmetry with respect to the point

(

2

,

2

3

)

. The valid angles of rotation for

F

are

120

∘

and

240

∘

. The boundary of

F

and the point

(

2

,

2

3

)

are shown in the following picture.

As a final example, the figure

{

(

x

,

y

)

:

-

3

≤

x

≤

-

1

⁢

and

⁢

(

x

+

1

)

2

+

y

2

≤

4

}

∪

(

[

-

1

,

1

]

×

[

-

2

,

2

]

)

∪

{

(

x

,

y

)

:

1

≤

x

≤

3

⁢

and

⁢

(

x

-

1

)

2

+

y

2

≤

4

}

is symmetric about the origin. The boundary of this figure and the point

(

0

,

0

)

are shown in the following picture.

If

E

⁢

(

F

)

=

F

and

E

is a

reflection

, then

F

has

reflectional symmetry

. In the special case that

V

=

ℝ

2

, the following terms are used:

•

F

has

line symmetry

;

•

F

has

symmetry about a line

;

•

F

is

symmetric about a line

.

The following are classic examples of line symmetry in

ℝ

2

:

•

Regular polygons: There are

n

lines of

symmetry

of a regular

n

-gon. Each of these

pass through

its center and at least one of its vertices.

•

Circles: A circle is symmetric about any line passing through its center.

As another example, the

isosceles trapezoid

defined by

T

=

{

(

x

,

y

)

:

0

≤

x

≤

6

⁢

and

⁢

0

≤

y

≤

min

⁡

{

x

,

2

,

-

x

+

6

}

}

is symmetric about

x

=

3

.

In the picture above, the boundary of

T

is drawn in black, and the line

x

=

3

is drawn in cyan.

Title

symmetry

Canonical name

Symmetry

Date of creation

2013-03-22 17:12:29

Last modified on

2013-03-22 17:12:29

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

18

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 51A10

Classification

msc 15A04

Classification

msc 51A15

Related topic

DihedralGroup

Related topic

DeterminingRotationsAndReflectionsInMathbbR2

Defines

symmetry about

Defines

symmetric

Defines

symmetric about

Defines

rotational symmetry

Defines

point symmetry

Defines

symmetry about a point

Defines

symmetric about a point

Defines

reflectional symmetry

Defines

line symmetry

Defines

symmetry about a line

Defines

symmetric about a line

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