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26 At Right Angles | Vol. 3, No. 1, March 2014

Invariance can also be seen in reflection. Figure

3 gives a design for the word “algebra” that is

invariant upon reflection, but with a twist. You

will notice that the left hand side is NOT the same

as the right hand side and yet the word is still

readable when reflected. So the invariance occurs

at the level of meaning even though the design is

not visually symmetric!

Figure 3. An ambigram for ‘algebra’ that remains

invariant on reflection. But is it really symmetric?

In this column, we use ambigrams to demonstrate

(and play with) mathematical ideas relating to

symmetry and invariance.

There are two common ways one encounters

symmetry in mathematics. The first is related to

graphs of equations in the coordinate plane, while

the other is related to symmetries of geometrical

objects, arising out of the Euclidean idea of

congruence. Let’s take each in turn.

Symmetries of a Graph

First let us examine the notions of symmetry

related to graphs of equations and functions.

All equations in x and y represent a relationship

between the two variables, which can be plotted

on a plane. A graph of an equation is a set of points

(x, y) which satisfy the equation. For example,

x2 + y2 = 1 represents the set of points at a distance

1 from the origin—i.e. it represents a circle.

Let (a, b) be a point in the first quadrant. Notice

that the point (–a, b) is the reflection of (a, b) in

the y-axis (See Figure 4a). Thus if a curve has the

property that (–x, y) lies on the curve whenever

(x, y) does, it is symmetric across the y-axis. Such

functions are known as even functions, probably

because y = x2

, y = x4

, y = x6

,... all have this

property. Figure 4b shows the graph of y = x2

; this

is an even function whose graph is a parabola.

Similarly, a curve is symmetrical across the origin

if it has the property that (–x, –y) lies on the curve

whenever (x, y) does. Functions whose graph

is of this kind are called odd functions, perhaps

because y = x, y = x3

, y = x5

,... all have this property.

See Figure 4c for an odd function.

A graph can also be symmetric across the x-axis.

Here (x, –y) lies on the curve whenever (x, y)

does. The graph of the equation x = y2

(another

parabola) is an example of such a graph. Can a

(real) function be symmetric across the x-axis?

Figure 5 shows a chain ambigram for “parabola”.

Compare the shape of this ambigram with the

graph in Figure 4b. The chain extends indefinitely,

just like the graph of the underlying equation!

Figure 5. A parabolic chain ambigram for “parabola”

Figure 4a. The point A (a, b)

and some symmetric points

Figure 4b. An even function:

y = x2

Figure 4c. An odd function: y = x3

Figure 2. An ambigram for “invariant”

that remains invariant on rotation

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At Right Angles | Vol. 3, No. 1, March 2014 27

The ambigram for “axis of symmetry” (Figure 6)

is symmetric across the y-axis. You can see a red y

as a part of the x in the middle. So this ambigram

displays symmetry across the y axis. At the same

time it is symmetric across the letter x!

Figure 6. The axis of symmetry: Is it the y-axis or the x?

Another possibility is to interchange the x and y

in an equation. Suppose the original curve is C1

and the one with x and y interchanged is C2. Thus

if (x, y) is a point on C1, then (y, x) lies on C2. By

looking at Figure 4a, convince yourself that the

point (b, a) is the reflection of (a, b) in the line y

= x, the straight line passing through the origin,

and inclined at an angle of 45°

to the positive side

of the x-axis. Thus the curve C2 is obtained from

C1 by reflection across the line y = x. If C1 and C2

(as above) are both graphs of functions, then

they are called inverse functions. An example of

such a pair: the exp (exponential y = ex

) and log

(logarithmic y = ln x) functions (Figure 7).

Figure 7. Inverse functions are symmetric across the

line y=x.

Figure 8 is a remarkable design that where exp

becomes log when reflected in the line y = x.

Figure 8: Exp becomes Log when reflected

across the diagonal line!

A great example of an inverse function is the

hyperbola y = 1 / x, defined for all non-zero real

numbers x (Figure 9). Its inverse is obtained

by interchanging x with y. But x = 1 / y can be

written y = 1 / x. So it is its own inverse, and thus

symmetric across the line y = x.

Figure 9. The symmetrical graph of the hyperbola. It is

its own inverse. And it’s odd, too!

of symmetry of symmetry

arithm

arithm

onential

onential

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28 At Right Angles | Vol. 3, No. 1, March 2014

The ambigram for “inverse” in Figure 10 is

inspired by the hyperbola.

Figure 10. An ambigram for the word “inverse”

shaped like a hyperbola.

It is symmetric across the origin and across the

line joining the two S's.

Seeking congruence

Another type of symmetry consideration arises

from the notion of congruence in plane geometry.

Two objects are considered to be congruent if one

object can be superposed on the other through

rotation, reflection and/or translation.

Figure 11 shows an ambigram of the word

“rotate”.

Figure 11. An ambigram for “rotate”. What happens

when you rotate it through 180°

?

This leads to the question: if we can rotate “rotate”

can we reflect “reflect”? Figure 12 is an ambigram

for “reflect” that is symmetric around the vertical

line in the middle.

Figure 12. An ambigram for “reflect.” What happens

when you hold it to a mirror?

Finally, the third operation is translation. An

example of this symmetry is shown by the chain

ambigram for sine in Figure 13.

Figure 13. A sine wave ambigram. It displays

translation symmetry.

The sine function satisfies many symmetry

properties. Perhaps the most important of them

is that it is periodic, i.e., if you shift (in other

words, translate) the functions by 2π, then you get

the same function back. In addition, it is an odd

function, and the ambigram is both periodic and

odd.

One can of course combine these transformations.

This is best understood by looking at the

symmetries of an equilateral triangle (see Figure

14) involving both rotation and reflection.

Figure 14. The 6 symmetries of an equilateral triangle.