Symmetry

A symmetry is a transformation that maps a figure onto itself as a whole. Individual points may move, but the complete set of points remains the same. That distinction separates “rotating a shape” from “a rotation symmetry of that shape.”

The object being considered must be specified. The boundary of a square has more symmetries than the same square with four different vertex labels or a coloured mark in one corner. Decorations and labels count when they are part of the figure.

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Reflection symmetry

A reflection axis divides a figure into parts that match under reflection. The matching points lie on perpendiculars to the axis at equal distances from it. Folding a paper model can suggest an axis, but the defining condition is the geometric correspondence.

An equilateral triangle has three reflection axes. A non-equilateral isosceles triangle has one, through the apex and the midpoint of the base. A scalene triangle has none.

A non-square rectangle has two reflection axes through opposite side midpoints. Its diagonals are not reflection axes because swapping length and width would not preserve the rectangle. A square has those two axes and both diagonals, giving four.

A non-square rectangle has two mirror axes

There are two reflection axes and rotational order 2. A quarter turn does not map this rectangle onto itself.

Diagram explanation and exact data

Horizontal and vertical reflections exchange matching corners. A 180° turn swaps opposite corners. A diagonal reflection or 90° turn exchanges long and short directions and fails for a non-square rectangle.

A non-square rectangle has two mirror axes: exact reference data
Transformation	Symmetry?
Horizontal reflection through C	Yes
Vertical reflection through C	Yes
Half turn about C	Yes
Quarter turn about C	No
Reflection in a diagonal	No

Rotational symmetry and order

A figure has a non-trivial rotational symmetry when a rotation smaller than a full turn maps it onto itself. For a finite rotational symmetry group, its order is the number of matching positions in one full revolution, including the identity position.

A regular pentagon has order $5$ and smallest positive symmetry angle $72^\circ$ . A non-square rectangle has order $2$ and smallest positive symmetry angle $180^\circ$ .

For order $n$ ,

\text{smallest positive symmetry angle}=\frac{360^\circ}{n}.

A figure with no non-trivial rotation has order $1$ under this convention. Do not call the smallest angle the order: $72^\circ$ is an angle, while $5$ is a count.

Symmetries of familiar figures

Symmetries of familiar figures: reference table
Unmarked figure	Reflection axes	Rotational order
Equilateral triangle	$3$	$3$
Non-equilateral isosceles triangle	$1$	$1$
Scalene triangle	$0$	$1$
Non-square rectangle	$2$	$2$
Non-square rhombus	$2$	$2$
General parallelogram that is neither rectangle nor rhombus	$0$	$2$
Square	$4$	$4$

A circle has infinitely many reflection axes, all through its centre, and every rotation about the centre preserves it. It does not have a smallest positive rotation angle, so assigning it an ordinary finite rotational order is inappropriate.

The word “general” in the parallelogram row matters. A statement about a generic member should not erase extra symmetries of its square or rhombus special cases.

Why regular polygons have predictable symmetry

The vertices of a regular $n$ -gon are equally spaced around its circumcircle. A turn of $360^\circ/n$ takes every vertex to the next one and every side to the next side, so it preserves the complete polygon.

For odd $n$ , each reflection axis joins a vertex to the opposite side midpoint. For even $n$ , there are $n/2$ axes through opposite vertices and $n/2$ through opposite side midpoints. Both cases give $n$ axes.

A regular decagon therefore has ten reflection axes and rotation order ten. Its smallest positive rotational symmetry is $36^\circ$ .

A coordinate way to test symmetry

For a finite point set, transform every point and check that the resulting set is the original set. Testing one pair is not enough.

The rectangle with vertices $(-3,-1)$ , $(3,-1)$ , $(3,1)$ and $(-3,1)$ is preserved by reflection in either coordinate axis. A $90^\circ$ rotation sends $(3,1)$ to $(-1,3)$ , which is not on this rectangle, so a quarter-turn is not a symmetry.

For a graph $y=f(x)$ , reflection in the $y$ -axis is a symmetry when $f(-x)=f(x)$ throughout a domain symmetric about zero. Such a function is even. The graph of $y=x^2$ satisfies this condition. Half-turn symmetry about the origin corresponds to $f(-x)=-f(x)$ on a symmetric domain; $y=x^3$ provides an example.

A function defined only for $x\ge0$ does not acquire a full $y$ -axis symmetry merely because its formula could be extended to an even function. The domain is part of the graph.

Repeating patterns and translation

An infinite repeating pattern can have translation symmetry. If shifting it by one repeat length leaves the complete pattern unchanged, that shift is a symmetry. A finite non-empty bounded figure cannot have a non-zero translation symmetry: repeatedly applying the translation would force points arbitrarily far from the original bounded region.

A glide reflection combines a reflection with a non-zero translation parallel to the mirror. It can preserve a repeating pattern even when neither component alone does. This introduces a broader class of pattern symmetries without changing the basic rule: the complete object must match itself after the transformation.

A cropped photograph of a repeating design may not retain the infinite pattern’s translation symmetry because the crop boundaries move. State whether the object is the ideal repeating pattern or the finite displayed piece.

Practice questions

Give the reflection-axis count and rotational order of a non-square rectangle.
A regular polygon has smallest positive rotational symmetry $45^\circ$ . How many sides does it have?
Does $y=x^2$ restricted to $x\ge0$ have reflection symmetry in the $y$ -axis?

Check your answers

It has two reflection axes and rotational order two. Its diagonals are not reflection axes.
$360/45=8$ , so it is a regular octagon.
No. Reflection takes positive- $x$ points to missing negative- $x$ points. Symmetry is a property of the graph with its actual domain, not the formula alone.

Sources and further reading

AMSI, Congruence: reflection and rotation of plane figures.
Euclid, Elements IV: regular figures and their geometric structure.