The best use of the mystery figure (in my opinion), and the answer to last month’s Geek Challenge is B: Used as an element of a proof of the Pythagorean Theorem. The figure generates an amazingly short derivation of the theorem, as our winner Gareth Meirion-Griffith of DMC demonstrates with just 2 lines of algebra:
Consider the figure above. The figure consists of:
1) A large square with sides of length C and an area of C2
2) Four right triangles with sides of length A, B, and C, and an area of (1/2)AB (four triangles combined give a total triangular area of 2AB)
3) A small square with sides of length B-A and an area of (B-A)2 = B2 + A2 - 2AB.
Thus, the area of the small square and the four triangles combined is:
Area = B2 + A2 -2AB + 2AB
Which simplifies to:
Area = B2 + A2
Thus, this proves that the entire area contained within the large square is equal to B2 + A2, which we also previously noted is equal to C2. Thus, B2 + A2 = C2, which is the Pythagorean theorem!
Similar solutions were also derived by Han Yang of DMC, and Adnaan Velji of DMC.
Now, to run down the rest of the answers:
A. It is the optimum target for focusing a monochrome camera
Not really. It would work, as would any sufficiently complex shape. A combination of straight and diagonal lines are nice to focus with, but it’s hardly optimum. A better focus target would have objects of many size scales, which this figure lacks. One really nice target to focus monochrome cameras with is the 1951 USAF resolution test chart:
C. It is used to design a centerless spiral staircase
With only four steps per revolution, the staircase would be way too steep. The spires of the Sagrada FamÃlia in Barcelona, Spain feature incredible centerless spiral staircases, but they are round.
D. It recognized as one of the most challenging shapes in Origami
An Honorable Mention is earned by Jordan Harris of DMC for his submission of the Origami Proof of the Pythagorean Theorem. It shows both that B is the correct answer, and that the figure is not that difficult to make in Origami.
E. There’s an even better use, not listed above.
A few people had fun with this one. Here are a couple of the submissions:
Furqan Ayub of DMC recognized the shape as a discrete element of the “four-bug problem” or Mice problem. In this problem, four bugs start at the corners of a square facing center, and each bug walks at constant velocity towards its neighboring bug to the right. The tracks the bugs make follow a spiral like this:
Chris Wagner analyzed the figure and concluded that the interior intersections always lie on semi-circles whose diameter are the outer edges of the square, and which intersect at the center. This shape represents the curve of the tooth path on a spiral bevel gear.
Send your comments to: geekchallenge@dmcinfo.com.