Stand a few inches away and look into one of the mirrors. Look at the spaces around that mirror.
With your finger close to one of the mirrors, what do you see in this mirror? What do you see in the surrounding mirrors?
Standing back a few feet, what patterns can you see in the collections of mirrors and the spaces between the mirrors?
Reflections in curved mirrors are different from those in flat mirrors. Each of the mirrors in this exhibit is a perfectly round shape called a sphere. The mirror surface is reflecting away from the center so it is a convex mirror.
The background behind the mirrors is black, so the area between spheres looks like a black triangle. Because of these triangles, the spheres in the center look like hexagons, six sided figures that have all sides the same length.
Many cars and trucks have side mirrors that are curved rather than flat. Women's makeup mirrors sometimes are curved mirrors. At hallway intersections in buildings and sometimes along a street you can see a convex mirror.
Have you identified the objects in this exhibit? You might already have spherical mirrors in your home - Christmas tree ornaments. Replacement plumbing waste pipe usually has a reflective surface and can be a curved cylindrical mirror.
Looking at the collection of spherical mirrors in this exhibit, how many different straight lines can you see?
How many different sizes of hexagons can you find?
How many different patterns can you find using the black triangles?
The spherical mirrors can be organized into three sets of straight lines. One set of lines are left to right or horizontal. A second set of lines are oriented between the upper left corner and lower right corner. The third set of lines are oriented between the upper right corner and lower left corner.
Each mirror looks like a small hexagon. A larger hexagon is made up of six mirrors, two on each side. A third hexagon is made up of three mirrors on each side.
A pair of black triangles looks like a bow tie. Six triangles together make a six sided star.
Choose any one of the black triangles. Combine it in turn with each of the black triangles around it. In each of these combinations a different corner of that triangle points towards the center of the bow tie. Why does each of these combinations still look like a bow tie?
The sides of the black triangle are all the same length, meaning the shape is an equilateral triangle.
How can you find the total number of mirrors without counting each of them?
The mirrors are arranged into rows. The odd number rows each have seven mirrors. The even number rows each have six mirrors. There are five rows of seven and four rows of six. The total number of mirrors is fifty-nine.
Honeybees build comb consisting of hexagons to store their honey. Pretend that the spherical mirrors in this exhibit are hexagonal storage areas and the black triangles are unusable wasted space. Is there any way to arrange these hexagons in a slightly different pattern to reduce the amount of wasted space?
The mirror hexagons in this exhibit share corners. Think of two hexagons instead that have a common side. Now add a third hexagon that shares sides with each of the first two. Keep adding hexagons in this way.
Why would hexagons be a good choice for the shape of honeycomb storage cells?
There is no unusable or wasted space between the cells.
House flies have compound eyes, meaning they have many individual "eyes" working together. Do all of the individual eyes see exactly the same thing?
Look at the image reflected by the different mirrors. Each has a slightly different image of the surrounding area. Imagine for a moment the amount of processing that must be performed to assemble the images from the fly's individual eyes into a unified picture.
challenge question 7 needed
challenge question 7 answer
Geometry, Concave, Convex, Reflection
further information listing needed
This exhibit is described in the Exploratorium Cookbook series.
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