Write the digit “1” exactly 317 times, and you get a palindromic prime number. Moreover, 317 itself is a prime number!

### Visual Proof (sum of cubes)

The sum of the sequence of the first *n* cubes equals [*n*(*n*+1)/2]² as shown below:

1³+2³+3³+…+*n*³ = (1+2+3+…+*n*)² = [*n*(*n*+1)/2]²

### 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

**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