Activity 11
Kaleidoscope Tessellations

Tessellations can be produced from polygons reflected in mirrors. In this activity we will examine the result of placing polygons inside folded mirrors and mirrors arranged as polygons.

Folded Mirrors

Using two rectangular mirrors, form a folded mirror by taping the two mirrors together with clear plastic tape on the front surfaces, as shown at the right. Place the folded mirror on the desktop so that the sides of the folded mirror lie on the labeled lines. Look into the center of the folded mirror and count the number of stars that appear. Write this information in the table below. Use a protractor to measure the mirror angle for each position and place this information in the table.

Position	A	B	C	D	E	F	G
Stars	_____	_____	_____	_____	_____	_____	_____
Angle	_____	_____	_____	_____	_____	_____	_____

What is the relationship between the number of stars that can be seen and the angle between the lines? If the angle were 20°, how many stars would you be able to see? If the angle were n°, how many stars? What would be the angle that would allow you to see 60 stars? What angle for n stars? Place the answers in the table below.

Angle	20°	n°	_____	_____
Stars	_____	_____	60	n

As Others See You

Place the folded mirror on the desktop so that you are looking directly into the center where the mirrors are joined. Adjust the angle between the mirrors to exactly 90°. When you look into the folded mirror, you will see yourself as others see you, rather than the way you see yourself in a mirror. Blink your left eye. Blink your right eye. A single mirror reverses the orientation of objects, but a double mirror, folded at exactly 90° reverses this orientation back to the original orientation. If the mirrors are not correctly aligned, the image will be distorted. While looking into the folded mirror, decrease the angle so that the image has only one eye. Now adjust the angle to show three eyes.

Regular Polygons

Each regular polygon can be formed using a folded mirror and a line segment. It is also possible to form a regular pentagon using two angled segments and a folded mirror. Place the hinge of the mirror at point P and open the mirror so that a portion of the segment stretches from one mirror to the other. Move the two sides of the folded mirror until you can see a regular hexagon.

What kind of triangle is formed by the mirrors and the line segment?_______________

Move the two sides until you can see a square.

What kind of triangle is formed by the mirrors and the line segment?_______________

Move the two sides until you can see a regular octagon.

What kind of triangle is formed by the mirrors and the line segment?_______________

The rays below form a 60° angle. If a folded mirror is placed on these rays, the segments will generate a regular polygon. Angle A will be an angle of this regular polygon. Use this result to guess the number of sides of the regular polygon that is generated. What would be the measure of angle B?

Determine the values of angles A, B and P needed to form regular polygons with 5 to 12 sides and place the results in the table below.

Sides	5	6	7	8	9	10	11	12
Angle A	_____	_____	_____	_____	_____	_____	_____	_____
Angle B	_____	_____	_____	_____	_____	_____	_____	_____
Angle P	_____	_____	_____	_____	_____	_____	_____	_____

Prism Kaleidoscopes

A prism kaleidoscope is formed by arranging mirrors into a polygonal shape. The polygon formed can be any triangle including equilateral, right isosceles, and scalene. The first prism kaleidoscope we will examine is in the shape of an equilateral triangle. Use the folded mirror for two sides and use a single mirror for the third side. Place these on the dotted triangular boundaries and look over the edge of a mirror to see the resulting tessellation formed by the dark lines inside the triangle. Write the code for the tessellation below the triangle. For the non-equilateral triangles, use the same approach.

Construct some of your own triangles so that the resulting tessellation is one of the regular or semi-regular tessellations. Color each portion with a different color so that the tessellation has each shape colored differently.