Exploration of Reflections

Back to Kaleidoscopes - Presentation Outline

Reverse Image

• parallel grid - to print out

• construct a non-symmetrical pattern block design in the middle of the parallel grid

• place a mirror (4"x6" are a nice size) on one of the heavy black lines to reflect the design

• construct the image (the reflection) with pattern blocks

Distance from figure to image

• find the distance from the figure (pattern block design) to the reflection line using the grid lines as a measurement tool

• find the distance from the reflection line to the image using the grid lines

• using the grid lines, find the distance from figure to image (should be twice the distance from the figure to the reflection line or from the reflection line to the image)

Parallel reflections = Translation

Mirrors

• place a mirror on each of the reflection lines, the mirrors should be facing each other - what do you see?

• construct as much as you can of the images; this overall design is known as a border or strip pattern, an infinite pattern in one direction

• locate two identical designs, not an image and its reflection but two identical designs; this relationship between these two images is called a slide or translation

• find the distance between these two identical images, the distance or amount of the translation; find the distance between the reflections - what is the relationship? (the translation is twice the distance between the reflections)

Miras (slow the process down to imagine what light is doing)

• parallel reflection - to print out

• using parallel lines A and B and a Mira, first reflect the arrow in A and draw the image

• turn the paper around and reflect the entire design in line B (this should create a reflection of the original arrow and the first image for a total of four arrows), draw the reflection

• continue the process (reflect in line A, reflect in line B) for as long as possible or desirable, the number of arrows should double with each reflection (bounce of light)

• the light bouncing between the mirrors, apparently instantaneously, produces an infinite design

Angled reflections = Rotation

Discuss

• Pattern block worksheets to print out -
  worksheet 1
  worksheet 2

• Student Activity: Pattern Block Flowers - using a single pattern block make a circular design (corresponding angles meet to form a "flower")

• there are two variations of this activity based on the ability/knowledge level of students - have the students measure each the angle on the pattern block and sum them to find that a circle has 360deg.; or knowing that a circle has 360deg., find the angle measure by dividing 360deg. by the number of pieces used (congruent central angles)

• emphasize the relationship between the size of the angle between the mirrors and the number of images produced in the design

• Additional Student Activity: Mirrors that Multiply from the AIMS Foundation, see bibliography

• Additional Student Activity: Symmetries with a Kaleidoscope, same principle but intended for an older audience

Angles - Mirrors

• angles - to print out

• place a small, non-symmetrical design of pattern blocks inside each angle

• place the hinged mirror (two mirrors taped together if nothing else) on the angle and determine the angle measure by counting the number of images (be sure to include the original design)

• each angle measure is a divisor of 360 to generate an integer number of images

Light - Miras (slow the process down to imagine what light is doing)

• using intersecting lines A and B and a Mira, reflect the arrow in line A and draw the image

• place the Mira on line B and reflect the entire design in line B, this includes a reflection of the original arrow and the first image; draw the images (should have a total of four arrows)

• reflect the arrows in line A again, while some arrows are overlaying existing ones this reflection should complete the circular design with two additional arrows; draw the images

• only three reflections were required to generate a complete kaleidoscopic design, further reflections (bounces of light) only continue to overlay existing images

Rotation - Miras

• in the completed circular design there are two triples of identical arrows that have three-fold (120deg.) symmetry

• if you consider a reflection in line A then line B, the resulting image (from the two reflections ignore the "intermediary" step) is a rotation of 120deg. from the original arrow where the distance of the rotation is twice the angle measure between line A and line B

• the rotation is in the same direction as the order of the reflections (counter-clockwise)

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