# Displays

There is so much to see and interact with at the Mathema Gallery. Below is a selection of displays that feature both a static and interactive element.

## Melencolia 1 & Magic Square

# Static

The name of this piece conjures initial thoughts of sadness. One interpretation is that Melencolia reflects the perpetually depressed mind of the mathematician, who can never fully succeed at perfecting his craft. However, it is not all doom and gloom!

Despite its name, Melencolia is a celebration of the hopeful quest for knowing. It is also a representation of pure mathematical genius. This is exhibited by a 4x4 'magic square' in the top right corner of the engraving, as well as a 3D shape now identified as a truncated triangular trapezohedron. **Can you spot both elements?**

# Interactive

What do you see when you look at this number puzzle? Do you see just a grid of numbers or something more? Now count the sum of the numbers in each row, column and diagonal. Can you see a pattern start to emerge? This is the beauty of the magic square.

One surprising fact about Durer's magic square is that there are more than 80 different geometric combinations that produce the magic sum of 34. **How many of them can you find?**

## The Interactive Board

# Static

In ancient times, it seems to have been understood that the side lengths of a right triangle could be related in the ratio of 3:4:5. The Greeks were also aware of this basic relationship but they developed it one step further.

They came up with a way to describe the relationship between the sides of all right triangles, not just specific ones. In other words, they were able to find other combinations of 3 numbers that were able to make right triangles. One of the most fascinating things about the Greek discovery is that it involved a proof which did not require algebra. Instead, the proof was entirely visual.

Today, there are more than 100 different proofs for this historic theorem, building on the foundation left by the Greeks thousands of years ago.

# Interactive

How did the Greeks prove their version of Pythagoras Theorem? Basically they started with a big square which was divided into two smaller squares of sides a and b respectively, and two rectangles with sides a and b; each rectangle was then split into two equal right triangles by drawing the diagonal c.

From there, the triangles were shifted to a different corner of the original square. The Pythagoreans used this rearrangement to show that the area of the square on the diagonal of each right triangle was simply the sum of the area occupied by the two smaller squares from the previous step.

Put this simple yet beautiful proof to the test by rearranging the pieces in the wood-piece shown at right. Follow the set of instructions above for further guidance.

## Pythagoras Theorem - Greek Interpretation

# Static

In ancient times, it seems to have been understood that the side lengths of a right triangle could be related in the ratio of 3:4:5. The Greeks were also aware of this basic relationship but they developed it one step further.

They came up with a way to describe the relationship between the sides of all right triangles, not just specific ones. In other words, they were able to find other combinations of 3 numbers that were able to make right triangles. One of the most fascinating things about the Greek discovery is that it involved a proof which did not require algebra. Instead, the proof was entirely visual.

Today, there are more than 100 different proofs for this historic theorem, building on the foundation left by the Greeks thousands of years ago.

# Interactive

How did the Greeks prove their version of Pythagoras Theorem? Basically they started with a big square which was divided into two smaller squares of sides a and b respectively, and two rectangles with sides a and b; each rectangle was then split into two equal right triangles by drawing the diagonal c.

From there, the triangles were shifted to a different corner of the original square. The Pythagoreans used this rearrangement to show that the area of the square on the diagonal of each right triangle was simply the sum of the area occupied by the two smaller squares from the previous step.

Put this simple yet beautiful proof to the test by rearranging the pieces in the wood-piece shown at right. Follow the set of instructions above for further guidance.

## Right Triangle Theorem - Chinese Interpretation

# Static

Despite bearing the name of a Greek mathematician, the history books suggest that Pythagoras' Theorem might have a separate origin. Take a look at the visual interpretation which came from an ancient Chinese text called The Nine Chapters on the Mathematical Art.

# Interactive

Much like the Greeks, the Chinese took 4 right triangles and arranged them around the edge of a grid so that the hypotenuse of each triangle became the side of a large square. The triangles were formed by drawing rectangles at each corner of the grid and then cutting them diagonally in half. For each triangle, the lengths of the shortest two sides were 3 and 4 tiles respectively.

To find the length of the hypotenuse it was necessary to know how many tiles made up the large square inside the grid. This could only be done by working out the total area occupied by right triangles and then subtracting this from the total number of tiles in the grid.

Once the area of the square was known, it was possible to work backwards to find not only the side length of that square but also (incidentally) the hypotenuse of each right triangle. In the case of the Chinese proof, the length of the hypotenuse (5 tiles) could be inferred from the fact that the square occupied an area equal to 25 tiles.

There is no direct evidence confirming that this was the method actually used by the Chinese. The only writing that was included with the image was the word "Behold"! The algebra that we associate with the theorem today was not invented until much later.

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