Defining Transformations

A transformation is an operation that moves a point or figure in a particular way. There are three basic transformations that moves a figure's position without changing its size or shape. These are called isomorphic transformations (iso = same; morph = shape). Each of the three basic operations is described below with a correct mathematical name, a nickname, and an associated "tool".

• Translation (Slide) > moves a figure along a given vector for a specific distance in a specific direction;
• Reflection (Flip) > reflects a figure across a given mirror or axis of reflection;
• Rotation (Spin) > rotates a figure around a given center for a specific angle of rotation.

These basic transformations can be used to create many basic polygons. For example, we can make a square starting with an isosceles right triangle in a variety of ways:

• we could translate the legs to make opposite sides of a square, or
• we could reflect across the hypotenuse to make a square, or
• we could rotate the hypotenuse (by 90°, centered at the right angle vertex); doing this four times will bring it all the way around.

It is the simplicity and symmetrical nature of the regular polygons that makes them so compelling and so important in both the arts and the sciences. Many of them can be constructed directly using compass and straight-edge, as well as using combinations of transformations. Some examples of construction strategies are given in the following sections.

Regular Triangle

Transformational Construction:

• Begin with a segment AB.
• Rotate the segment through 60° in opposite directions (clockwise and counter-clockwise) about each of the endpoints.

Euclidean Construction:

• Let the symbol "O(A, AB)" represent "the circle with center A and radius AB".
• Use the circle tool to construct O(A, AB)
• Construct O(B, AB).
• Shift/select the points A and B; Display >> Color these points. De-select the points, then change the color back to black.
• Shift/select the two circles; Construct >> Points of intersection.
• Connect A, B and either of the points of intersection to form a triangle. How would you confirm that this triangle is regular or not?

Square

Transformational Construction:

• Begin with a segment AB.
• Rotate the segment 90° clockwise, centered about endpoint A.
• Connect to form the segment joining B to it's image, B'.
• Reflect AB and AB' over the mirror BB'.

Euclidean Construction:

• Construct a pair of perpendicular lines, intersecting at point A.
• Let the symbol "O(A, AB)" represent "the circle with center A and radius AB". Construct O(A, AB) so that B is a point on one of the lines.
• Locate the other three points of intersection of the circle with the lines.
• Connect the four points of intersection to construct a square with center A and control point B.

Other Regular Polygons

Can you describe at least one transformational and one Euclidean construction strategy for the regular pentagon? regular octagon?

Are there patterns governing these strategies?

Parallelogram

Transformational Construction:

• Begin with a segment AB.
• Place a point C somewhere near point A.
• Connect C and A to form segment AC.
• Translate segment AB along vector AC.
• Translate segment AC along vector AB.

Euclidean Construction:

• Construct segment AB and a point C near point A. Construct segment AC.
• Construct the line parallel to AB passing through C.
• Construct the line parallel to AC passing through B.
• The point of intersection of the constructed parallels, point D, provides the fourth vertex of a parallelogram ABDC.

Other Types of Polygons

Can you describe at least one transformational and one Euclidean construction strategy for:

… an isosceles triangle?
… right triangles?
… a rhombus?
… a kite?
… an equilateral but non-regular pentagon?
… an equiangular but non-regular hexagon?
… a figure that exhibits rotational symmetry but not line symmetry?
… a figure that exhibits line symmetry but not rotational symmetry?

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