The answers to this question are pretty funny, and also fairly informative!

What Is an Example of a Counterintuitive Mathematical Result?

The Hydra Game will always lead to a result that you probably wouldn’t expect.

This is an
aptly named blog. Check out all the answers to this question on Quora for more counterintuitive results!

The Dragon Curve

This is a fractal resembles a dragon and is also known as the Jurassic Park Fractal.
Starting from a base segment, replace each segment by 2 segments with a right angle and with a rotation of 45° alternatively to the right and to the left. Continue this with the newly generated segments.

Fathom the Universe

consider a large rectangle R which is tiled with small rectangles of various shapes(but with sides vertical or horizontal), each of which possess the following property P: ” one of the sides of the rectangle has integer length.”

prove the rectangle R has itself has the property P.

Each dot is only moving in a straight line, but is created by balls moving in circles through 3 dimensional space.

[Click for more interesting science facts and gifs]

This is the magic hexagon. The numbers add up to 38 along each straight line. It can be called the magic hexagon rather than a magic hexagon because there are no other hexagons numbered 1,2,…,n with this property, no matter how many layers the arrangement has (except for the one which is just one hexagon with 1 written in it, but that’s hardly magical…).

(Source: Mathematical Gems I by Ross Honsberger)

Two curves cut all circles at right angles: straight line and a tractrix.

I didn’t know what a tractrix was, so I Googled it! Tractrix definition from Wikipedia: Tractrix (from the Latin verb trahere ”pull, drag”; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed

Here’s a gif from Wikipedia showing a tractrix being created from dragging a pole:

Using the chain rule is like peeling an onion. You have to deal with every layer at a time and if it’s too big you’ll start crying.
 — Calculus professor (via mathprofessorquotes)

timb:

How a hole is drilled to be made square, the red shape in the center would be the cutting tool.

it shares the same principle as a Reuleaux triangle but with one rounded corner so that the cut square does not have rounded edges; the cutting tool follows the path of the rounded edge that is tangent to the sides of the outer square. because a tangent line is perpendicular to the radius, as the cutting tool follows the path of the rounded edge it turns precisely 90° to create a sharp edged perfect square.

Good, I always thought he didn’t have enough imagination for mathematics.
 — David Hilbert, upon hearing one of his students stopped attending his lectures, in favor of poetry. (via curiosamathematica)
Kaprekar number

A Kaprekar number is a number k such that if you square k, and then add the n rightmost digits to the n or n-1 leftmost digits, you yield k.

Let k = 9.
92 = 81. 1+8 = 9.

Let k = 45.
452 = 2025. Since k had two digits, 4 and 5 respectively, n = 2, so you would add the n rightmost digits to the remaining leftmost digits. 20 + 25 = 45.

Let k = 703.
732 = 494209. n = 3. 494 + 209 = 703.

Absolute zero is no longer absolute zero

Scientists have rewritten the known laws of physics after hitting a temperature lower than absolute zero. Physicists at the Ludwig Maximilian University in Germany created a quantum gas using potassium atoms, fixing them in a standard lattice group using magnetic fields and lasers.

http://www.nature.com/news/quantum-gas-goes-below-absolute-zero-1.12146

That article explains more in depth. A gas created with potassium atoms was held in a lattice structure with lasers and magnetic fields. The atoms were at their lowest energy and most stable state repelling one another. By adjusting the magnetic fields, the physicists made the atoms attract one another. This put the atoms in their highest energy state, but the physicists simultaneously adjusted the lasers so that the structure of atoms wouldn’t collapse inwards. This caused the temperature of the gas to transition from just above absolute zero (0 K) to a few billionths of a Kelvin below absolute zero.

Basically, high energy states that were difficult or impossible to achieve at positive temperatures became stable at negative temperatures.

Still don’t get it. I mean temperature is essentially a measurement of energy right? So if something is in very fast motion, it is very hot, but if something is not in motion at all, and maybe has no volume(?) then it is at the theoretical point 0K, no? So how can something be moving less than not at all? How can something have less volume than nothing? Maybe I just need to take a couple physics courses before I can understand this…

It might be helpful to note that in terms of “hotness” and “coldness” the temperature scale actually runs from

0K—> 100K —-> positive infinity —-> negative infinity —-> -100K —-> -0K

So a temperature in negative Kelvin is technically “hotter” than any positive Kelvin value (the inverse temperature scale make a bit more logical sense as it runs -infinity to +infinity).

This all comes from the relation between energy and entropy and is all good fun but is still something I don’t fully understand (and that’s after two semesters of grad statistical mechanics).

Was thinking about this “article” more last night and just wanted to clarify that the original link (which is just a tiny blurb really) is highly sensationalized. The nature link provided by shaliomar is a much better reference.

The concept of negative temperatures on the Kelvin scale has been around at least since the 1950s, so rest assured that the known laws of physics have not in any way been rewritten just because a research group was able to arrange an experimental system to temporarily be in a negative temperature configuration (it’s not the first time it’s been done but it is certainly some very interesting research! Please do give the nature link a read.)

However, I wanted to point out that negative temperatures are not achieved by cooling something below absolute zero. Absolute zero is defined such that it is the “coldest” temperature (minimized entropy). Instead, the researchers used the experimental techniques outlined in the nature link to arrange their system in such a way that the majority of particles were in high energy states. This corresponds to a negative temperature system.

Why?

In all systems, the particles are distributed among any available energy states. For positive temperature systems, the majority of particles are in low energy states. Adding energy to the system increases the number of available configurations for the particles (increase in entropy) because more particles are entering the less densely populated higher energy levels (particles going from high population density levels to lower population density levels). For negative temperature systems, the majority of particles would instead be in the higher energy states (this is why negative Kelvin values are actually “hotter” than any positive Kelvin value). Now, adding energy to the system decreases the number of available configurations (decrease in entropy) because particles are being removed from energy levels that are lowly populated and put into energy levels that are already highly populated.

In other words, in statistical mechanics, positive temperature means that adding energy increases entropy while negative temperature means that adding energy decreases entropy.

tl;dr The laws of physics are fine just really weird