Cubes from all Angles

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Tetrahedral kites by Alexander Graham Bell, (1898-1912)

Can You Flatten a Sphere?

Can you flatten a sphere?

The answer is NO, you can not. This is why all map projections are innacurate and distorted, requiring some form of compromise between how accurate the angles, distances and areas in a globe are represented.

This is all due to Gauss’s Theorema Egregium, which dictates that you can only bend surfaces without distortion/stretching if you don’t change their Gaussian curvature.

The Gaussian curvature is an intrinsic and important property of a surface. Planes, cylinders and cones all have zero Gaussian curvature, and this is why you can make a tube or a party hat out of a flat piece of paper. A sphere has a positive Gaussian curvature, and a saddle shape has a negative one, so you cannot make those starting out with something flat.

If you like pizza then you are probably intimately familiar with this theorem. That universal trick of bending a pizza slice so it stiffens up is a direct result of the theorem, as the bend forces the other direction to stay flat as to maintain zero Gaussian curvature on the slice. Here’s a Numberphile video explaining it in more detail.

However, there are several ways to approximate a sphere as a collection of shapes you can flatten. For instance, you can project the surface of the sphere onto an icosahedron, a solid with 20 equal triangular faces, giving you what it is called the Dymaxion projection.

The Dymaxion map projection.

The problem with this technique is that you still have a sphere approximated by flat shapes, and not curved ones.

One of the earliest proofs of the surface area of the sphere (4πr²) came from the great Greek mathematician Archimedes. He realized that he could approximate the surface of the sphere arbitrarily close by stacks of truncated cones. The animation below shows this construction.

The great thing about cones is that not only they are curved surfaces, they also have zero curvature! This means we can flatten each of those conical strips onto a flat sheet of paper, which will then be a good approximation of a sphere.

So what does this flattened sphere approximated by conical strips look like? Check the image below.

But this is not the only way to distribute the strips. We could also align them by a corner, like this:

All of this is not exactly new, of course. In the limit, what you have is called a American polyconic projection, which does require stretching in order to fill the gaps between the ending of the strips. Gauss’s Theorema Egregium demands this.

But I never saw anyone assembling one of these polyconic approximations. I wanted to try it out with paper, and that photo above is the result.

It’s really hard to put together and it doesn’t hold itself up too well, but it’s a nice little reminder that math works after all!

Here’s the PDF to print it out, if you want to try it yourself.

All from 1ucasvb

Warped 2D Grids

"Made out of thousands of tiny little triangles, these delicately thatched composites strike a pleasing visual chord through their careful balance of negative and positive space. Created by LA-based artist Katy Ann Gilmore, the organic geometric images are realized through a variety of influences including mathematics, nature, and visual art. The tiny latched shapes spread outward until final pieces emerge, looking like hand-crafted topographic maps.

Gilmore studied mathematics and visual arts, and the experience gained in each field shines through in her unique style. Her layered, two-dimensional drawings explore the concept of perpendicular planes and the distortion of three-dimensional space. Some areas of the images recede where others spring forward, creating the illusion of a finely woven net gripping the land's curvature.

Gilmore’s breathtaking work was initially inspired by a curiosity for how 2D grids would look when warped in all directions, and how the fluctuation of empty space around physical objects affects our perception of solidity. Her work draws thousands of tiny elements together to form beautiful realizations of mathematical concepts through the application of an artistic eye."

mymodernmet

 

 

Beads + Patterns = Art

This installation took Liza Lou more than 5 years,and 30 million beads to create.

Her intricate bead patterns are a fantastic example of geometry and repeated patterns.

photo source

Fruit Fractions

These images are great prompts for a maths session on fractions and patterns.

"Turkish artist Şakir Gökçebağ creates new geometric shapes by cutting into various fruits and vegetables. Each redefined shape in his series of food art is crafted with precision and aligned accordingly. The arrangements created by Gökçebağ present interesting patterns that are, no pun intended, a feast for the eyes!"

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Tile the Plane

Big math news! It’s been thirty years since mathematicians last found a convex pentagon that could “tile the plane.” The latest discovery (by Jennifer McLoud-Mann, Casey Mann, and David Von Derau) was published earlier this month. Full story.

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Fractal Designs

from mathmajik.tumblr.com:

"Fascinated by the intricate patterns formed by fractals, basically math processes that repeat incessantly in an ongoing feedback loop, UK physicist-turned-web developer Tom Beddard makes impossibly elaborate complexes that look like they belong in the gritty cities of a dystopian fantasy. Basically what he does is write and run programs on his computer that spit out patterns—"the best outcomes are often the least expected!“ he writes—that he in turn massages (by way of shadowing and the like) into looking like faceted steel-and-concrete architecture.

Sources:
Curbed.com
Architizer.com"