Transforming maps is a fun way to experiment with geometry. You can twist and turn maps, combine them with other maps, and cut them up and reassemble them-the possibilities are endless!

Many people have experimented with transforming maps, but there are few rules that dictate how you can transform a map. These rules make it more geometrically creative.

The first rule is that the *map must* be connected. You cannot cut up a map into islands or *separate pieces unless* you are working with a geography map where this is appropriate.

The second rule is that the map must be complete. You cannot have blank spaces or missing landmasses on your *map unless* you are working with a geography map where this is appropriate.

The third rule is that the dimensions of the *map must remain* the same. The dimensions of the land do not change, nor does the size of the country on the map.

## Map △b to △c

Now let’s look at the other direction: from △a”b”c to △abc. There are **two possible transformations** that could have occurred to get from one map to the other.

The first possibility is that △b could have mapped to △c. In other words, the *triangle edge b could* have shifted down and to the right, landing on c. The second possibility is that both *edges b* and c could have shifted down and to the right, meeting at a point d.

Check out these animations to see how this would have happened!

Both of these possibilities show that there are multiple ways that individual elements of a shape can shift, which can make it hard to identify whether or not an element has moved until after it has done so.

## Map △a to △b”

In this case, the triangle that is mapped is a right triangle, and the letter that is mapped to itself is b. Since the letter b is in the middle of the left side of the map △a and the right side of △a”, there are only two possible transformations that could have occurred.

One *possible transformation would* be a reflection over the line △b”. This would result in a map △ab” of △a”b”c” onto itself, which does not look very similar. The other possible transformation would be a rotation by 180 degrees about the line △b”. This would result in a map △abc of △a onto itself, which looks more similar.

These are not the only possible transformations that could have occurred, but they are some common ones to think about.

## Map △b to △c”

Given the **fundamental group structure** of the square lattice, there are a finite number of *possible lattice maps*. These include rotations, translations, and reflections.

Rotations represent a 360° turn, or equivalently a *point moving halfway around* the square. A rotation of 90° represents a quarter turn, or a *point moving one*-quarter of the way around the square.

A translation is a displacement of the entire square. A translation of one unit to the right represents a new position of the square one unit to the right compared to the original position.

A reflection represents an exchange of left and right sides of the square. A reflection over the axis that passes through points (0, 0) and (1, 1) represents such an exchange.

All these transformations are binary functions: they take only two points as input and output only two points.

## Combine maps

Another way to find new maps is to combine existing maps. The map you get as a result will depend on how you combine the maps.

If you combine two maps that relate the same points, then you can merge the maps by mapping the points from one map to the other. This would result in a single map with fewer points.

If you *combine two different sets* of points, then you can create a new map by applying both maps to one set of points and then linking those points with those from the other map. This would result in **two separate maps**.

These methods can be applied to any kind of map, not just geographic ones! For example, you **could take two recipes** and mash them together to get a new recipe. Or you **could take different pieces** of art and merge them together to get a new piece of art.

## Repeat!

The repeat transformation is one of the most common and basic translations. It repeats a segment of the line as it moves each point to its new position.

As an example, imagine that you are mapping a line segment from point A to point B. When you trace the path of the line, you find that it *curves slightly outward*. To accurately reflect this curving path, you have to draw a slightly longer line from A to B.

This would be the case for any curve: The higher number of points on the outward part of the curve, the more points must be repeated on the inside of the new shape that is formed.

Repeat transformations are used in **almost every type** of map projection. Because they **keep lines straight**, they are helpful in *illustrating geographic features* such as borders and coastlines.

## alt=”user-uploaded image” src=”https://i.imgur.com/0zF1BVJ.png”>

There are **infinitely many permutations** of transformations that could have occurred to map △abc to △a”b”c”. Some of these are more physically plausible than others.

For example, you could not have had a rotation occur before the translation. This would result in a different shape, and our goal is to keep the same shape.

You also could not have had a reflection occur before the translation or the dilation. This would also result in a different shape, and our goal is to keep the same shape.

Another possible combination is having a **shear transformation occur** before the dilation. This would result in a different shape, and our goal is to keep the same shape.

The only valid combinations are having a translation, dilation, or *scaling occur first*, then a rotation.

## transformations needed are actually very simple, but the number of possible combinations is huge! Imagine that you had all the possible maps of triples of points (which would be quite a lot!), and imagine how many combinations you could get just by combining two maps at a time and keeping one fixed. The number of possible transformations is much, much bigger than that because we’re also keeping all three points fixed at once.

First, imagine that your map only had ** two points**, instead of three. In this case, the only

**be to flip the map over. Since there are only two points on the map, there is only one side to flip it over to show the other side.**

**possible transformation would**Second, imagine that your map only had one point. In this case, the only possible transformation would be to fold up the map. Since there is only one point on the map, there is only one direction to fold it in.

## are infinitely many ways for this transformation to occur; however, there is a very large set of transformations that will map onto . The set consists precisely of those transformations for which each line has length 1.

The special class of transformations that preserve length is called linear transformations. A linear transformation between two vectors and is a function *f: V \rightarrow V* such that

for all *v_1, v_2 \in V*. The last condition says that the vector you get when you apply the transformation to the vector is also in the original space.

Linear transformations play an important role in geometry, and we will return to them later.