Hi there,

This is a question I recently received from a geometry math student, but I didn't know the answer:

What is a "mira," and how is it used in terms of topology (in geometry)?  If you can answer my question, and even refer me to online or other resources to find out more, I would greatly appreciate it.

Thanks

Constantinos

Hi Constantinos

A MIRA is the the brand name for a specific plastic tool for exploration / construction with reflections.

It is made out of a special red plastic, so that it BOTH reflects what is in front AND lets you see through to what is behind.

So, for example, you can 'see' if there is a reflective symmetry, by placing this on the object and comparing the reflected image with the back image. If they coincide, there is mirror symmetry. You can check the symmetries of a line - by seeing which placements (e.g. perpendicular to the line at any point) place the image and the extension of the line on top of one another.

You can ALSO use it to construct right bisectors of line segments or angles, or most of the constuctions you might do with origami. (I.e. all the constructions which can be done with straightedge and compass - and more.)

The original MIRA is produced in Willowdale (now part of the Northern Section of Toronto). It has two supporting end pieces which let it stand easily - but make it harder to butt one up against another to compare the product of two reflections. There is a competitor which has only one end with a cross piece for support - and therefore less plastics (and cheaper). Both are available from teacher supply places (like Spectrum Education near Toronto).

I have even manufactured a cheap 'mira' for use on a sphere to test the symmetries of a 'line' (great circle) on a sphere, and duplicate some of the constructions on a sphere. This can be done with the plastic 'rolodex' covers - put a blue one inside the red one. If gives much the same colour and effect - a mix of reflection and transparency which lets you compare the reflection with the extended original.

So they are used in schools which explore transformations in elementary school. Also used in some mathematics education methods classes - and even in my third year geometry class here at York! (We concentrate on isometries and transformations. After all, groups of transformations are the very definition of a geometry in the modern - post 1870 world!)

Walter Whiteley
Geometer
York University

Hi Constantinos

I tried to find a diagram of a MIRA for you. The best I found was on a book advertisement

Harley
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