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Chapter 6: Problem 261

Another view of the dihedral group of the square is that it is the group of all distance preserving functions, also called isometries, from a square to itself. Notice that an isometry must be a bijection. Any rigid motion of the square preserves the distances between all points of the square. However, it is conceivable that there might be some isometries that do not arise from rigid motions. (We will see some later on in the case of a cube.) Show that there are exactly eight isometires (distance preserving functions) from a square to itself. (h)

Short Answer

Expert verified
There are exactly 8 distance-preserving functions (isometries) for a square.

Step by step solution

01

Understand the Problem

Understand the requirement to show that there are exactly eight distance-preserving functions (isometries) for a square. The square maintains the distances between all points.
02

Identify Rigid Motions

Identify all rigid motions of the square including rotations and reflections. Rigid motions preserve the shape and distance within the geometric figure.
03

List All Rotations

List the possible rotations: 0 degrees (identity), 90 degrees, 180 degrees, and 270 degrees. These rotations map the square onto itself while preserving distances.
04

List All Reflections

List the possible reflections: reflection over the vertical axis, horizontal axis, and the two diagonal axes. These reflections also map the square onto itself while preserving distances.
05

Count Total Isometries

Count the total number of isometries by adding the four rotations and four reflections: 4 + 4 = 8.
06

Conclusion

Conclude that there are exactly eight distinct isometries (distance-preserving functions) from the square to itself.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

isometries
Isometries are functions that preserve distances between points within a geometric figure. In the context of a square, isometries ensure that the shape and relative positioning of all points within the square remain exactly the same. Therefore, if two points on the square were 5 units apart before applying an isometry, they will still be 5 units apart afterward. Isometries are crucial in geometry because they allow transformations that do not alter the fundamental properties of shapes and figures.
In our case, the dihedral group of the square is defined by all the distance-preserving functions (isometries) from a square to itself.
rigid motions
Rigid motions are transformations that preserve the size and shape of a geometric object. Common examples of rigid motions include rotations and reflections. By definition, rigid motions do not deform the figures to which they are applied.
For the square, we can identify the main rigid motions:
  • Rotations (0 degrees, 90 degrees, 180 degrees, and 270 degrees)
  • Reflections (over the vertical, horizontal, and diagonal axes)
All these movements map the square onto itself while preserving distances between points, thereby demonstrating they are both rigid and isometric.
distance-preserving functions
Distance-preserving functions, or isometries, are essential in understanding geometric transformations because they ensure the integrity of the shapes involved. When a function is distance-preserving, it means that the distance between any two points remains constant before and after the transformation.
Mathematically, if the distance between two points, say A and B, is represented as d(A, B), a distance-preserving function f maintains that d(f(A), f(B)) = d(A, B). This is why all rotations and reflections of a square are concluded to be distance-preserving.
geometric transformations
Geometric transformations involve moving or changing a figure while preserving certain properties. They include translating, rotating, reflecting, and scaling. However, within the context of isometries and rigid motions, we're focusing on transformations that do not change the size or shape of the figure.
For a square, the primary geometric transformations that are considered isometries are:
  • Rotations: 0 degrees (identity rotation), 90 degrees, 180 degrees, and 270 degrees.
  • Reflections: over the vertical axis, horizontal axis, and the diagonals of the square.
These transformations collectively form the dihedral group of the square, showing all eight possible isometries.

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Most popular questions from this chapter

Another relation that you may have learned about in school, perhaps in the guise of "clock arithmetic," is the relation of equivalence modulo \(n\). For integers (positive, negative, or zero) \(a\) and \(b\), we write \(a \equiv b\) \((\bmod n)\) to mean that \(a-b\) is an integer multiple of \(n,\) and in this case, we say that \(a\) is congruent to \(b\) modulo \(n\) and write \(a \equiv b(\bmod n) . .\) Show that the relation of congruence modulo \(n\) is an equivalence relation.

If \(f: S \rightarrow T\) is a function, we say that \(f \operatorname{maps} x\) to \(y\) as another way to say that \(f(x)=y .\) Suppose \(S=T=\\{1,2,3\\} .\) Give a function from \(S\) to \(T\) that is not onto. Notice that two different members of \(S\) have mapped to the same element of \(T\). Thus when we say that \(f\) associates one and only one element of \(T\) to each element of \(S,\) it is quite possible that the one and only one element \(f(1)\) that \(f\) maps 1 to is exactly the same as the one and only one element \(f(2)\) that \(f\) maps 2 to.

Can you find subgroups of the symmetric group \(S_{4}\) with two elements? Three elements? Four elements? Six elements? (For each positive answer, describe a subgroup. For each negative answer, explain why not.)

Suppose \(P=\left\\{S_{1}, S_{2}, S_{3}, \ldots, S_{k}\right\\}\) is a partition of \(S .\) Define two elements of \(S\) to be related if they are in the same set \(S_{i},\) and otherwise not to be related. Show that this relation is an equivalence relation. Show that the equivalence classes of the equivalence relation are the sets \(S_{i}\).

Digraphs of functions help us to visualize whether or not they are onto or one-to-one. For example, let both \(S\) and \(T\) be the set \\{-2,-1,0,1,2\\} and let \(S^{\prime}\) and \(T^{\prime}\) be the set \(\\{0,1,2\\} .\) Let \(f(x)=2-|x|\) (a) Draw the digraph of the function \(f\) assuming its domain is \(S\) and its range is \(T\). Use the digraph to explain why or why not this function \(\operatorname{maps} S\) onto \(T\) (b) Use the digraph of the previous part to explain whether or not the function is one-to one. (c) Draw the digraph of the function \(f\) assuming its domain is \(S\) and its range is \(T^{\prime} .\) Use the digraph to explain whether or not the function is onto. (d) Use the digraph of the previous part to explain whether or not the function is one-to-one. (e) Draw the digraph of the function \(f\) assuming its domain is \(S^{\prime}\) and its range is \(T^{\prime} .\) Use the digraph to explain whether the function is onto. (f) Use the digraph of the previous part to explain whether the function is one-to-one. (g) Suppose the function \(f\) has domain \(S^{\prime}\) and range \(T\). Draw the digraph of \(f\) and use it to explain whether \(f\) is onto. (h) Use the digraph of the previous part to explain whether \(f\) is one-toone.
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