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

**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.

Interested in my 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”

### A curious right triangle

The sides of a pentagon, hexagon, and decagon, inscribed in congruent circles, form a right triangle.

### Humor: Apple Pi

### The Arithmetic-Geometric Mean Inequality

A visual intuitive proof that **√ab** cannot be larger than **(a+b)/2**, where a, b ∈ R*+