Actually, string theory is something completely different, but it’s a cute title for this geometry problem at Futility Closet. A boy has his toy boat in the water, and he is pulling it to shore by a string. If he pulls in one yard of string, will the boat advance a yard, or less than a yard, or more than a yard? The answer may surprise you. Link -via TYWKIWDBI

Photo: Nick Sayers [Flickr]

Behold the Hyperbolic Coffee Cactus, created by Nick Sayers out of "630 coffee stirrers, drilled with 2,520 holes, pinned together with 1,260 half cocktail sticks. All by hand."

George Hart of Make Magazine explains:

Nick Sayers enjoys making geometric constructions from unusual materials. Here’s an organic-looking sculpture he made from 630 coffee stirrers, with “blobs” protruding in the twelve directions of a dodecahedron‘s faces. [...]

Each “blob” form is based on half an icosidodecahedron, with small triangles surrounded by three large pentagons.

Link

Best answer to a geometry quiz question ever. It would almost be worth missing the question just to use this clever answer.

Link via Geekologie

Talk about a twist- Steve Kass made noodles that are Möbius strips! You can see them cooked and ready for dinner at Flicker. Link -via Evil Mad Linkblog

(Image credit: Flickr user Steve Kass)

You can consider Aakash Nihalani the MacGyver of urban art. All he’s ever needed to create his wondeful artwork are paper, tape, cardboard and a little bit of geometry.

More at his official website [warning, Flash] or Flickr page – via Unurth

Yellow Galaxy

Infinity Triangle

Link – via unurth street art

Update 9/27/10 – edited to correct the artists involved.

Sculpture

Sculptor Bathsheba Grossman creates metal and crystal artworks of forms found in math, physics, biology, and astronomy. Grossmen shows us Borromean rings, hypercubes, gyroids, fractals, Calabi-Yau spaces, and interlaced sculptures based on the five Platonic solids. I particularly like this Voronoi network wrapped onto a Möbius toroid, sculpted in white glass.

Grossman created this beautiful lamp from one of her Ora series sculptures. Available in several lamp styles from Materialise.

Jewelry

The Julia set is a fractal equation that produces a series of rather pleasing spirals. Designer Marc Newson took that fractal shape and designed a necklace of 2,000 diamonds and sapphires that took jewelry craftsmen 1,500 hours to put together. Note that the necklace is not symmetrical, but still has a sense of balance. See how the jeweler, Boucheron, advertises the necklace.

Drawing

Probably the best known artist to use math concepts in his works is M.C. Escher. Many of his 2-dimensional drawings turned 3-dimensional geometry on its head. The lithograph titled Waterfall illustrates the concept of the Penrose triangle, also called the impossible triangle. Escher also explored tessellations in many of his drawings.

Computer Imaging

Paul Nylander was one of the developers of the Mandelbulb that we saw in a previous math post. He is a computer engineer and an artist who renders math and science concepts into colorful images including animated .gifs to help us visualize their 3- or 4-dimensional structures. Shown is a Dodeca-Spidroball, a variation on the spidron, which was invented by Daniel Erdely in 1979.

Belgian mechanical engineer Jos Leys renders and animates all kinds of math concepts into beautiful forms that boggle the mind. His artworks include fractals, Kleinian groups, inversive geometry, recursions, tessellations, knots, and tilings in both images and video renderings to show 3- and 4-dimensional effects. The image above is called Indra200, an example of “Kleinian jewelry“.

Other artists rendering math images worth checking out include Torolf Sauermann, Brian Johnston, Mehrdad Garousi, and the late Titia Van Beugen.

Video

(YouTube link)

Creating visual representations of math concepts became easier with computer rendering software and digital video capabilities. That doesn’t mean it is simple. Homporgo, the artist who created this video of a Mandelbox zoom said in a comment:

Believe me Bill, I wanted to go further too, but at the end part a single frame took 18 minutes to render, and the whole 1:27 minute video needed 12 days nonstop rendering. I felt thats more than enough at the time.

Twelve days! The result looks worth it to me. How about you? See more fractals on video in this post.

Previously at Neatorama: A Non-Math Look at Math Objects and A Non-Math Look at Math Shapes.

Hyperboloid

What mathematicians call a hyperboloid of one sheet is a really cool structure that is made up of many (actually an infinite number) of perfectly straight lines that look to us like a curved structure. First, imagine that you have a cube. Stand it on one of its corners and spin it like a top, then look at it from the side -the sides seem to be curved, but you know they aren’t. Now, take a handful of uncooked spaghetti noodles. Use two hands, and twist the strands loosely. It forms the shape of a hyperboloid structure, which looks like a cooling tower at a nuclear reactor. All the spaghetti noodles are still straight, but the shape of the handful is curved. In architecture, this idea enables builders to produce curved structures by using straight line supports.

Apollonian Gasket

An Apollonian gasket is a fractal generated when you mash as many round soap bubbles together as you can. At least, that’s what it looks like. The pattern is based on threes: every circle touches two other circles. As you add more circles in the smaller spaces, they also touch two existing circles (and eventually many smaller ones). The number of smaller circles that can be added is mathematically infinite. Frothing soap bubbles can help us picture the Appolonian gasket, but the analogy is flawed, because real world soap bubbles do not like empty spaces. There is a limit to the volume of soap, and surface tension will connect round bubbles and flatten them against each other. This fractal is named for the ancient Greek mathematician Apollonius of Perga. The 3-dimensional fractal of this sort is called the Apollonian sphere packing, which is a pretty descriptive name for a math term.

Möbius Strip

When I was very young, first or second grade, my father told me he could make a piece of paper with only one side. Then he took a strip of paper, gave one end a half-twist, and taped the ends together. Then he showed me how he could draw one line down the strip without stopping and it covered the whole strip, no matter which way you turned it! I couldn’t wait to show the Möbius strip to my friends at school. When I did, they just stared at me and told me I was weird.

Maybe “continuous plane” is a better description than “one sided paper”. The Möbius strip is named for August Ferdinand Möbius who discovered it in 1858. Johann Benedict Listing also came up with the concept around the same time and actually published his work, but maybe someone thought calling it a “Listing strip” would be confusing. Anyway, the Möbius strip does have some real-world applications. For example, conveyor belts and recording tapes with a half-twist last twice as long as they would otherwise because the entire surface is used instead of just one side of a two-sided strip. It’s also an attention-getter in art and even architecture.

Klein Bottle

The Klein bottle came about from an attempt to make a 3-dimensional Möbius strip. In 1882, mathematician Felix Klein theorized about a container that had no inside or outside -just one “side”- in this manner:

Take a rectangle and join one pair of opposite sides — you’ll now have a cylinder. Now join the other pair of sides with a half-twist. That last step isn’t possible in our universe, sad to say. A true Klein Bottle requires 4-dimensions because the surface has to pass through itself without a hole.

Here in the real (3-dimensional) world, we cheat by passing the neck of the bottle through itself using a real hole. You can see that process in sequence. You can see (and purchase) the finished product. You can put water into a Klein bottle (carefully), but you can’t see a defined lip, meaning there is no discrete border between what is the inside and what is the outside of the container.

The Heliospheric Current Sheet

The Heliospheric current sheet is the shape of the sun’s magnetic field. As the sun rotates, the magnetic field is forced into a spiral, called a Parker Spiral. The spiral has four arms with two different polar phases. As these rotate, the shape of the current sheet (which is the effect of the magnetic field on the plasma in the solar wind) forms a 3-dimensional spiral that resembles a twirling dancer with a voluminous skirt. Picture the bottom polarity as the dancer pushing her skirt down (with two hands) while the force of the twirl raises it like the upper polarity does. The two different polarities causes the vertical shape and the outward spiral causes the horizontal shape. You can reproduce this shape by twirling around while holding a gushing garden hose. Be sure to move the hose up and down as you turn -or better yet, watch while someone else does all this.

Vesica Piscis

The words vesica piscis literally translates to English as fish bladder. It is the name of the particular shape created when two circles of the same size overlap so that the edge of each circle touches the center of the other, such as the shape you see in the middle part of a two-circle Venn diagram. You could easily say this is just two identical curves placed together, but those curves have a lot of symbolism, going back to ancient times. The two curves with a bit of the crossover left on in the shape of a fish were used as a symbol of Christianity in its early days (and today as well). The shape is also the basis of the Gothic arch, which is stronger than a regular round arch in that it resists pressure from the sides as well as from above. It enabled architects to build huge structures with taller but stable doors and windows. The vesica piscis is also symbolic of the overlapping visual field of our two eyes -and, of course, some see it as a vagina.

Buddhabrot

Pareidolia is the tendency of humans to see meaningful shapes in everyday objects, like the image of the Virgin Mary in a grilled cheese sandwich. In studying the Mandelbrot set of fractals, which we looked at the in the previous post, several people noticed that if you turned the traditional rending of the fractal on its side, the shape looks like a sitting Buddha. This led to the search for the precise formula to generate a Buddhabrot. Turn your head to left while looking at this image and try to tell me you don’t see a golden statue. The number of iterations (how many fractal levels are rendered to be visible) has a lot to do with the shape you see.

As in the previous post, you can get a much more mathematical explanation of each of these shapes, including formulas, by clicking on the links in this post.

Superegg

(Image credit:  Sir48 at da.wikipedia)

A superegg is a mathematical shape constructed by rotating a superellipse around an axis to the formula of |x/a|2.5 + |y/b|2.5 = 1, where a/b = 4/3. (If you search for “superegg formula”, you are liable to find something completely different.) But you don’t want to bother with formulas, do you? Just look at it! From the side, the superegg looks a bit like a cylinder, but has no corners. If you cut one horizontally, the cross-section will be a circle. However, unlike a natural egg, you can stand the superegg on its end -either end, as a matter of fact, as it is vertically as well as horizontally symmetrical, although it has no straight lines that you can find -although the curvature is zero at the ends, the “ends” are actually quite small and appear to be rounded. The superegg was popularized by Danish mathematician and physicist Piet Hein, who used the shape in designs for household items such as furniture, ice cubes, and candles, as well as a novelty toy (sometimes referred to as a stress-reliever) by itself.

Torus

I learned about the torus from crossword puzzles. If the clue says “donut shape”, the answer is torus. The solid is produced by rotating a circle around an imaginary axis, but in this surface of revolution, the axis is outside the circle. The resulting shape is a ring torus. Other torus shapes are produced when the axis is touching or slightly inside the circle. Some really strange mathematical shapes are produced when the rotating plane of the circle is not quite round, or is itself rotating around a point in the plane.  A toroid is a ring or donut shaped solid produced by a surface of revolution not necessarily limited to a circle. For example, a square used in this manner will produce a ring that would be uncomfortable on your finger. A toroidal polyhedron is a torus constructed with or converted into flat surfaces, with the shape dependent on how many flat surfaces you use. Toroidal Polyhedron would be a cool name for a band.

Gömböc

You might remember Weebles -they wobble, but they don’t fall down. However, if the heavy weight in the bottom of the toy ever came loose, you had a Weeble that fell down. In 1995, Russian mathematician Vladimir Arnold questioned whether there could be a 3-dimensional shape that would always return to its original position without the help of internal weights. If a shape could be found that had as few as two points of equilibrium, one stable and one unstable, the shape would naturally return to balancing on the one stable point. For a long time, mathematicians thought the shape was impossible. But in 2006, Gábor Domokos and Péter Várkonyi developed the gömböc. This odd shape has only two points it could possibly balance upon, and the point on top is too “pointed” to be stable. So, if you roll a gömböc around, it will soon right itself, returning to an upright position because of its shape, not because of any internal irregularities. It’s a Weeble that doesn’t wear out! Objet Geometries made the first fabricated gömböcs. They were numbered as a limited series (inside, using transparent materials of the same density as the rest of the object) and professor Arnold was presented with number one. You can buy one of your own.

Spidron

A spidron sounds like something that spins, and that’s what it looks like as well. A spidron is defined as a two-dimensional figure “consisting of an alternating sequence of equilateral and isosceles (30°, 30°, 120°) triangles.” OK, you know an equilateral triangle has three sides all the same length. An isosceles triangle doesn’t (the hypotenuse is longer). So when the equilateral triangle adjoins the right side of an triangle isosceles on one side and the hypotenuse on the other side, you see that the size of the isosceles triangles get smaller as they go on -and therefore the size of the equilateral triangle decreases as well. So the shape peters out as it grows longer, and it shifts a bit to the side because of the isosceles triangle shape. Since the shape grows in at least two directions (or it’s only a semi-spidron), the shape looks like a twisty pinwheel blade of sorts. But what good is it? Just take a look at how they fit together!

There are more spidron designs than can be contained in one post. See, with enough spidrons, any math whiz can be another M.C. Escher!

Bernoulli’s Spiral

Jacob Bernoulli was a Swiss mathematician who studied spirals produced by logarithms, which he called Spira mirabilis, or “the marvelous spiral”. These are often called Bernoulli’s spirals now in honor of the man who popularized them. There is a mathematical formula, but the simple explanation is that the spiral grows larger as it progresses outward, but the curve does not. This gives it a particular shape you may recognize in nature, most commonly in snail shells and seashells. You can also see logarithmic spirals in weather systems, spiral galaxies, and in flowers, in which case there are often spirals growing in opposite directions.

Mandelbrot Fractals

(Image credit: Wolfgang Beyer)

The word fractal has the same root as fracture, meaning broken. Fractals are geometric shapes that, if broken into parts, fracture into parts that are a smaller version of the whole (the same shape, that is) which is called a self-similar image. So those parts themselves are copies of the whole, and can be further broken down into smaller parts that are also copies of the whole. We non-math folks sometimes call this recursion. True fractals don’t exist in nature, as there is a limit to how small finite units can go -as far as we know, atoms only come in certain shapes. However, in geometry they can be broken down infinitely. A Mandelbrot set (named after mathematician Benoît Mandelbrot, who coined the term fractal) is a complex set of points on a plane that has a fractal boundary. In other words, the shape of the set repeats itself along the edge, and each repeating shape also contains repeating shapes, no matter how closely you zoom into the plane. This is easier to understand by looking at it.

(YouTube link)

That’s pretty mind-blowing by itself. Could a Mandelbrot set be made into a 3-dimensional shape?

Mandelbulb

The Mandelbulb is a 3-dimensional representation of a Mandelbrot set. Daniel White set out to go beyond the idea of rotating a plane around an imaginary axis to create a 3-dimensional shape. He published an experimental formula to create a true 3D Mandelbrot set in 2007. Paul Nylander from Fractal Forums helped refine the formula. The resulting shape called a Mandelbulb is not quite what they are looking for -yet, but the two men are still working on perfecting the formula. See more pictures of the Mandelbulb here. No, I can’t even pretend to know how they accomplished that one, so with my mind adequately blown, I’ll end this list.

There are formulas for each of these shapes, which mean little if you aren’t at ease with higher geometry. If you are interested, you can find those formulas by following the links in this post.

San Francisco-based artist/coder Robert Hodgin of Flight 404 Blog created some of the most mesmerizing sculptures using magnetized balls and cylinders. They’re part of the Gray Area Foundation for the Arts (GAFFTA) exhibition.

That has got to take some mad skillz because I can easily envision the whole thing collapsing into a pile of magnetized blob at the slightest touch.

MAKE Blog has the gallery: Link | Robert’s official webpage

Related: Buckyballs over at the NeatoShop

An object or scene in the visual world is projected as a two-dimensional image on the retina of each eye, so what we see can also be treated as flat sheet: the visual field. Every point on this sheet can be pin-pointed by two coordinates, just like a point on a map, or a point on the flat model of V1. The alternating regions of light and dark that make up a geometric hallucination are caused by alternating regions of high and low neural activity in V1 — regions where the neurons are firing very rapidly and regions where they are not firing rapidly.

A closer look at the types of specialized neurons in the V1 field and how they interact with each other explains the geometric patterns.

Bressloff and his colleagues used a generalised version of the equations from the original model to let the system evolve. The result was a model that is not only more accurate in terms of the anatomy of V1, but can also generate geometric patterns in the visual field that the original model was unable to produce. These include lattice tunnels, honeycombs and cobwebs that are better characterised in terms of the orientation of contours within them, than in terms of contrasting regions of light and dark.

That’s about as simple as I can make it in a short blurb; the entire article explains it better. Yes, there is math involved. Link -via Metafilter

Found at Cliff Pickover’s always excellent Reality Carnival. It took me a while to get it!

Previously on Neatorama: The Math Book: Milestones in the History of Math