Take any triangle and divide it into sub-triangles as shown. Inscribe a circle in each of these smaller triangles.
Rearrange the order of the circles and adjust the intervening lines so that each line touches two circles:
No matter how this is done, each circle will always fit perfectly in its triangle. Here are some proofs.
“There exist only two kinds of modern mathematics books: ones which you cannot read beyond the first page and ones which you cannot read beyond the first sentence.” — Physics Nobelist Yang Chen-Ning
A self-inventorying crossword from Lee Sallows:
This one even tallies the number of spaces it contains!
(Thanks, Lee!)
In 1798, Adrien-Marie Legendre claimed that 6 is not the sum of two rational cubes.
In 1865, Gabriel Lamé observed that 6 = (37/21)3 + (17/21)3.
There’s no blue circle here. The space among the lines is white. In the presence of black lines, the hue of a colored object seems to bleed into the surrounding background.
The phenomenon was first discovered in 1971. It’s known as neon color spreading.
This Alternative Heritage plaque adorns the house in Hull where John Venn was born in 1834.
A segment of an image tends to seem larger when it’s filled with visual elements. This is true whether the elements are discrete or continuous.
Above, the right part of the top figure and the left part of the bottom figure each seems to fill more than half of its tier, though plainly these impressions can’t both be valid.
The illusion was first studied by German physicists Johann Joseph Oppel and August Kundt in the 1860s.
In all three of these figures, the gray ring’s color is uniform. But in the second and third figures the upper half appears distinctly darker, showing that the brain exaggerates differences in brightness between adjacent surfaces. German psychologist Kurt Koffka first reported the effect in 1935.
In the Saturday Review in 1946, architect Gerald Lynton Kaufman proposed nine “poetry forms in mathematics”:
Famously, in a group of 23 randomly chosen people, the chance is slightly higher than 50 percent that two will share a birthday.
Here’s an interesting variant: If the group consists of an equal number of girls and boys, what’s the minimum size at which it’s probable that a girl and a boy share a birthday?
Surprisingly, the answer is only 32 (16 girls and 16 boys). If we want a girl and a boy to share the same birth month, we need only 6 children before this becomes probable.
(Tony Crilly and Shekhar Nandy, “The Birthday Problem for Boys and Girls,” Mathematical Gazette 71:455 [March 1987], 19-22. See Shared Birthdays.)
