Wrinkled Doughnut Solves Geometrical Mystery

This may be the weirdest doughnut you have ever seen, but it solves a long-standing geometrical puzzle that evaded mathematicians including Nobel laureate John Nash, who inspired the film A Beautiful Mind.

1. ___________ According to its rules, a certain type of flat square - in which opposite edges have been mathematically linked - is equivalent to a holed-doughnut, or torus, because one can easily be turned into the other. First, form a cylinder by joining the top edge of the square to the bottom edge, then bend that cylinder into a circle and join its two open ends.

There is just one problem: for the two ends to meet, the torus must be stretched in a way that distorts the original shape of the square. Any horizontal lines on the original square will be stretched on the torus, while vertical lines will remain the same. (2. _________ They are forced then to make compromises such as inflating the size of Greenland, which can appear similar in size to Africa on standard maps but is actually one-fourteenth as big.)

But could there be an alternative torus that leaves both horizontal and vertical line lengths unchanged? In the 1950s, game theorist and economist John Nash, together with mathematician Nicolaas Kuiper, proved that such a torus could exist. 3. __________ As a result, no one knew what it would look like. "It's like describing a cooking recipe at the molecular level," says Francis Lazarus at the University of Grenoble in France.

Now, Lazarus and a team of mathematicians from Grenoble and the University of Lyon have managed to visualise the shape of this torus. Starting with a shrunken version of the regular, smooth torus, they wrinkle the surface in the horizontal direction, increasing the length of just the vertical lines. They then apply further wrinkles in other directions until the lengths of both vertical and horizontal lines are equal to the lengths of these lines on the square. 4. __________.

The method of wrinkling is known as convex integration theory. 5. _________. He says the theory could now be applied to solving complicated systems of equations that arise from problems in physics and biology. Meanwhile, he and his colleagues plan to bring their strange new shape into the real world using a 3D printer.

I. Use the sentences (A – E) to fill in the gaps (1 – 5). Explain your choice.

A. Cartographers encounter a similar problem when unwrapping a globe of the Earth to create flat maps.

B. The result is the bizarre-looking torus that is pictured above-right.

C.Topology is the branch of mathematics concerned with the geometric deformations of objects.

D."Until now people though it was a very complicated technique," says Lazarus. "This work proves you can actually use it."

E. However, their methods only worked at a tiny scale, making it too difficult to actually visualise the shape.

II. A. Find the words from the article corresponding to the definition:

1. __________ existing or in effect for a long time;

2. __________ to get away from or avoid (imprisonment, captors, etc); escape;

3. __________ a ring-shaped surface generated by rotating a circle about a coplanar line that does not intersect the circle;

4. __________ to twist or pull out of shape; make bent or misshapen; contort; deform;

5. __________ to form a mental image of (something incapable of being viewed or not at that moment visible);

6. __________ a mathematical statement that two expressions are equal: it is either an identity in which the variables can assume any value, or a conditional equation in which the variables have only certain values (roots);

7. __________ to cause to increase excessively; puff up; swell

8. __________ to come upon or meet casually or unexpectedly; to be faced with;

II. B. Use the words from II. A to fill in the gaps.

- You can use the same word twice.

- You might need to change the part of speech to fit the word into the gap.

a. A dense ring of gas and dust – called a _____ – girdles and obscures the dying star and contains most of the star's ejected gas.

b. Night-vision goggles used by the pilots, he testified, were less than perfect because they can _____ gun flashes and tracer bullets.

c. He tried to ______ these molecules, the proteins bent and folded into patterns of hideous complexity.

d. The jury found that she had advised some clients to ______ tax.

e. An ______ is balanced if there are the same numbers of each type of atom on each side of the _______.

f. We can speculate that the reason for this omission was a _______ Western prejudice against the concept.

g. There was little doubt Stand Aside was destined to sink at some stage and the decision was made to _______ the life rafts.

h. The spacecraft will now be able to send back pictures of the rings and other scientific data collected in its close _______ with Saturn.

III. Say if the statements below are true or false:

- In theory, there is a special type of flat square that can be turned into a torus.

- The distortion of the original shape is clearly seen by the way Greenland and Africa are shown on geographical maps.

- J. Nash and N. Kuiper have failed to provide a visual model of the torus.

- To visualize the shape the researchers had to wrinkle the surface in the vertical direction.

- Convex integration theory can only be applied to topology.

IV. Think of the answers to the following questions:

1. What type of flat square is equivalent to a holed doughnut?

2. What exactly prevents the two ends to meet?

3. What problem did the researchers encounter back in the 1950s?

4. How did the team of mathematicians manage to visualize the shape of the torus?

5. Speak in more detail about convex integration theory. What are its possible applications?

V. Summarise the article in 5 sentences.

VI. Speak on the part of the Grenoble team and describe the way the researchers have managed to solve the puzzle.

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