‘Deliverance 2’, 2013
Earth As Art by NASA
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The 3-dimensional toroid maze is actually made of a 2-dimensional maze wrapped up together. When this is done, some interesting mathematical properties occur.
The donut is actually made from a two-dimensional maze, such as the one with the solution depicted in red. Mazes such as this one have two important properties. One, it is possible to reach every single point in the maze from any other point. Two, if you pick any two points, there is exactly one unique path that connects the two without backtracking. Together, these characteristics make what mathematicians call a “spanning tree.”
When a maze like this is made into a cylinder, and the ends of that cylinder are connected, a “maze donut” is born.
“There are a huge number of these mazes that can be made, and a very simple algorithm can create each maze,” said Russ Lyons, Professor of Mathematics at Indiana University. “I wish I had the time to solve each one.
According to Lyons, although there is a fuzzy, practical significance to this area of mathematics in chemistry (polymers) and physics (two-dimensional quantum gravity), the math behind the donut is studied mostly out of sheer human curiosity.
And it is that curiosity that leads him one step further into the maze, where there are several more interesting properties.
Of all the mazes that can be created, the shortest possible solution is one that travels in two straight lines: one along the left side and one along the bottom. The longest possible solution is one in which the red line visits every single place in the grid.
If the maze is made in a 10-by-10 grid, the shortest solution is 20 lines long (10-by-2) and the longest is 100 (10-by-10). But odds against randomly creating either of these mazes exceed imagination. So if left to chance, about how long will the solution to the maze be? Recent mathematical proofs have shown that for very large mazes, the average solution will be about 1.9 x n^1.25, where “n” is how long each side of the grid is. So for a 10-by-10 grid like the one shown above, “n” is 10, and the average solution is 34 lines long.
Recently, Lyons took the problem another step further. He connected the top and bottom of the maze to make a cylinder (like the one seen below), and then connected the ends to make a donut. When this is done, the previously mentioned characteristics disappear. But others take their place.
This is a three-dimensional representation of a simple, 2-dimensional maze wrapped into a cylinder. The next step is to connect the ends, making a donut.
Now, instead of one unique path connecting any two points, there is a unique set of edges that traverse the donut in any direction. That is, once on the purple path shown, one can go around and around the donut as many times as one wants without backtracking. The maze below shows this set of lines on a two-dimensional grid. However, if the path is blocked, and if one is not allowed to go once around the donut via this path, it again becomes a spanning tree, meaning there is only one way to get from point A to point B.
This is the 2-dimensional version of the 3-dimensional “maze donut” shown above.
In a randomly generated donut, how many edges will likely be included in the purple path that spans the donut? Nobody is quite sure, but Lyons has reason to believe it is again 1.9 x n^1.25. This is just one characteristic of this problem that is currently being explored.
High resolution digital terrain models, or DTMs, are topographic maps of Mars as imaged by the High-Resolution Imaging Science Experiment (HiRISE) on board NASA’s Mars Reconnaissance Orbiter (MRO). They are created by grabbing two separate images of the same region of the Martian surface during different orbits. These “stereo pairs” (with different angles of inclination) are used to precisely measure the elevation of the landscape after a complex and time consuming series of steps including calibration by mission scientists and calculations by a powerful algorithm. The result is nothing short of beautiful. So get immersed in Mars’ technicolor landscape and see the scale of some of those awesome geological formations on the Red (blue, green and yellow) Planet. View the entire gallery…
Saturn, photographed by Cassini, 20-21 July 2009. The two large moons in the top-left are Enceladus and Tethys; Mimas makes a fleeting appearance in the lower-right; and if you watch carefully, you can see Epimetheus and Janus just outside the F Ring.
‘Charged charmonium’ confounds particle physicists
Physicists working independently at two different particle-physics labs have found tantalizing evidence for a new and mysterious hadron. Dubbed Zc(3900), the particle seems to be a “charged charmonium” and is made from quarks assembled in a way that has possibly never been seen before. Further studies of Zc(3900) could provide important new information about the strong force that glues together quarks in hadrons.
The prime spiral, also known as Ulam’s spiral, is a plot in which the positive integers are arranged in a spiral with primes indicated in some way along the spiral. Unexpected patterns of diagonal lines are apparent in such a plot. This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam’s slightly more formal prose, it “appears to exhibit a strongly nonrandom appearance”
In the above variation of the Ulam spiral, red squares represent prime numbers and white squares represent non-primes. Image source.
Mauro Savoldi - “Johnny Hermann is the alter-ego of the craftsman and designer Mauro Savoldi from Milan. He re-creates the vibrant, colorful magic of summer ices in objects of minimal design, recalling one of the sweetest and most nostalgic treasures of our past. The original popsicle was invented by an 11-year-old boy in San Francisco in 1905 – and by a strange coincidence it was piece of wood that made the whole story possible! Childhood memories and fresh emotions are fused in the shape and materials of these creations.”