The ways ravens carry two donuts at once! Read more.
Topology & Ethology
Equable Triangles
There are only five integer-sided triangles whose area is numerically equal to its perimeter:
(5, 12, 13), (6, 8, 10), (6, 25, 29), (7, 15, 20), and (9, 10, 17)
As you can see from the picture, only 2 of them are right triangles.
Euler’s Line
“Euler’s line” (red) is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).
Other notable points that lie on the Euler line include the de Longchamps point, the Schiffler point, the Exeter point, and the Gossard perspector.
Trefoil Klein Bottle
The “Klein Bottle” is what happens when you merge two “Möbius Strips” together: the resulting shape will still have only one side – with its inside and outside merging into one. Obectively, such a paradoxical shape is clearly not possible within our 3-D reality and requires a fourth dimensional jump at some point to make it all come together. Also, because true Klein bottles do not have discernible “inside” or “outside”, they have ZERO VOLUME. As a result, these objects can only be simulated as an “impossible art” in our world, or only modeled with a “fake” 3-D intersection, instead of a true extra-dimensional joint. There are a lot of Klein Bottle model variants, this one is the most intriguing.
Elementary 4-manifold topology
Cut any of the loops of strings below and the whole chain comes apart.
You will find many of these kinds of puzzles in my book “Impossible Folding Puzzles and Other Mathematical Paradoxes” .
Thébault’s theorem
If you place squares on the sides of any parallelogram, their centers will always form a square.
Geometric Illusion: Vertigo Pattern
Do you feel queasy when you look at this wallpaper? Though they appear to be sloped, the columns of stacked white and black patterns are perfectly straight and PARALLEL to each other.
This optical illusion is available as prints and posters at our online gallery.
Interested in optical illusions? Feel free to visit my author page.
Coxeter Disc
Infinite flavor in a finite fruit pastry space!
Further reading: http://www.ams.org/publicoutreach/feature-column/fcarc-circle-limit
Infinite Pythagorean Triplets
Consider the following simple progression of whole and fractional numbers (with odd denominators):
1 1/3, 2 2/5, 3 3/7, 4 4/9, 5 5/11, 6 6/13, 7 7/15, 8 8/17, 9 9/19, …
Any term of this progression can produce a Pythagorean triplet, for instance:
4 4/9 = 40/9; the numbers 40 and 9 are the sides of a right triangle, and the hypotenuse is one greater than the largest side (40 + 1 = 41).
Golden Ratio (And Its Inverse) In Yin Yang
The philosophy of the Yin Yang is depicted by the the “taichi symbol” (taijitu). In fact, Yin Yang is a concept of dualism, describing how seemingly opposite or contrary forces may actually be complementary,
Curiously enough, in the taichi symbol are hidden the golden ratio and its reverse. As shown in the picture. Continue reading “Golden Ratio (And Its Inverse) In Yin Yang”