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Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

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Look Before You Leap

Can you spot a cunning way to work out the missing length?

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Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

It Depends on Your Point of View!

Age 14 to 16 Challenge Level:

Why do this problem?

This problem offers an opportunity for clear cross-curricular work between maths and art. The mathematics behind the creation of anamorphic art is intriguing, and yet requires nothing more complicated than some simple ratio and similar triangle work.

Possible approach

Images of pavement art or advertising from sporting venues provide a "hook" to grab the attention of learners and offer a genuine real-life context for exploring similar triangles.

It would be great to begin by sharing lots of examples of anamorphic art with learners. This link has lots of fantastic examples of pavement art, but images from sports, the elongated bike painted on a cycle path, or fine art such as Holbein's "The Ambassadors" in the problem can also be used to stimulate discussion about this technique.

Here are the sorts of ideas that it might be useful to discuss:
Looking at an anamorphic image from the "wrong" place, what do you notice? (For example, here and here.)
Usually, when I view an object lying flat on a table, the parts further away from me appear smaller. How can I make it appear as if the object is standing up (so all points are the same distance away from my eye)?

Once learners have had a chance to explore the ideas behind anamorphic art informally, set them the challenge of creating an image of their own. To start, take a square grid and use similar triangles to calculate how the grid needs to be transformed to be visible from their chosen distance and height. Once they have worked out how to transform a grid, they can transfer an image from the square grid to the anamorphic grid to create a picture that appears to be standing.

Key questions

Looking at an anamorphic image from the "wrong" place, what do you notice?
How can I trick the viewer into believing that an object is standing up when in fact they are looking at a flat image?

Possible extension

Try out these techniques on a large scale using playground chalks on the school yard.

Possible support

Start by transforming a single small square viewed from a particular point, and build this up to creating a square grid.