Snow sculptures from Breckenridge

Lately, we published snow sculpture by Bathsheba Grossman which was represented at Breckenridge snow sculpture contest. Here some other mathematical snow sculptures represented at Breckenridge in other years.

Knot divided (2005)

This is a triply twisted Moebius band. There is a self-referential beauty in our sculpture: If one forms a Moebius band by twisting a belt through three half-turns (instead of just one), then the band's edge forms a trefoil knot.

Whirled White Web (2003)

This sculpture is a 3-fold symmetrical whirl of twisted and stretched saddle shapes. Such saddles occur naturally in soap films that are spanning warped wire frames; such "minimal surfaces" are nature's way of creating strength in delicate structures. Our sculpture uses these natural ideas to create an intricate network of ribs and internal spaces suspended from a web of three mutually interwoven double loops.

Turning a Snowball Inside Out (2004)

In the 1960s, mathematicians showed how to turn a sphere inside out. To do this, the surface must pass through itself, but no tears or creases are allowed to form at any point. This design is an artistic interpretation of the halfway point of such a sphere-everting process, where the surface displays half of its outside, shown as a solid form, as well as half of its inside, rendered as a transparent grid.

Here we see a snail shell but it's also a representation of fractal twisting to it's center.

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This article was created by materials of
http://www.cs.berkeley.edu/~sequin/SCULPTS/sculpts.html

Graffiti in Barcelona

Graffiti in Barcelona
Rosa y Dany

In this wonderful illustration we see the same artistic effect which Escher used in his lithograph "Reptiles" where reptiles go from tessellated surface of table into three-dimensional environment, crawl by things on the table and returns into tessellated surface. Unlike, Escher's artwork here reptiles appears from mosaic.

Fractals in nature

Fractals are not only mathematical abstraction, which can be used for creation of abstract or realistic images (see images by Keith Mackay). They can be found in our world in various plants, shells etc. Below you can see several examples of fractals in nature.

The most known fractal is fern leaf. Every little leaf of fern repeats whole large leaf.

Another interesting nature fractal is romanesco cauliflower, which is a cross between broccoli and Cauliflower, which accentuates the great fractal spiral patterns on the top.
In cactus below leafs become twisted towards the center.

Fractals also exists in micro world. Viruses, mould and bacterial aggregate colonies spontaneously assume fractal shapes.

The best known fractal representation among animals is nautilus shell.
Lightning Strikes and Electrical Discharge creates fractal formations called Lichtenberg figures on rocks, grass,wood or even people!
Many symmetrical wonderful fractal shapes can be found in snowflakes.

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This post was created by materials of
http://www.its.caltech.edu/~atomic/snowcrystals/
http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html

Temperal paradox

Temperal Paradox
Patrick's Picks

After mystery of blank spot in Escher's lithograph "Print Gallery" was revealed many derived images have appeared. Very often these images are created from real photos as it was done in Stanford University with photo of Hendrik Lenstra who revealed the mystery.

The image above was created by Patricks Picks from photo of clock face. He also created many other similar images. You can see them in his profile on Flickr or in Escher's Droste Print Gallery pool.

Dog Dream

Dog Dream

In this artwork created by Kaz Maslanka we can see artistic representation of fractal like Koch snowflake.

The inscription at the top is in Korean:

Dog Dream = Irrationality / Importance

Jenn 3D

Images above and on the right were created using computer program Jenn 3D which can draw graphs of finite Coxeter groups on four generators. Jenn builds the graphs using the Todd-Coxeter algorithm, embeds them into the 3-sphere, and stereographically projects them onto euclidean 3-space. The models really live in the hypersphere so they looked curved in our flat space.

Jenn 3D is licenced under version 2 of GNU public license. You can download executable file from developer's site. Unfortunately, developer does not provide executable for Windows platform at present, but he says you can download the source and compile it yourself.

Cubic houses in Rotterdam

A set of innovating houses was built in 1984 on Overblaak Street in Rotterdam in The Netherlands. These houses have shape of cube which were tilted to 45 degrees. The composition consists of 32 cubes, all attached to each other. The houses contain three floors.

This set of buildings was designed by architect Piet Blom. He is best known for his 'Kubuswonigen' buildings built in Helmond and Rotterdam.

Cube houses have interecting feature. They looks like impossible figure from bird's-eye view as it seen in Google Maps. Special covered roofs of cubes looks like snow-covered mountain peaks.

Fractals by Jock Cooper

A very impressive collection of fractals with more than one thousand image was created by Jock Cooper. Very bright and colourful images of Julia sets are represented in the main gallery. Besides traditional types of fractals, you can find three-dimensional mechanistic fractals in special section.

Fractal art by Keith Mackay

Many knows about simple IFS fractal that depicts fern leaf. It shows the complexity of simple forms. The large fern consists of a number of similar ferns smaller size and so on.

Maybe, this simple image of fern shows us secrets of nature. Trees, leafs, corals and other objects can be depicted using IFS fractals. So we can create our own world where all things will be fractals of various forms.

Some artists creates realistic images using fractals. One of them is Keith Mackay. He creates landscapes, underwater spaces and creatures. It's incredible that some images looks amazingly realistic. His birds and fishes looks artificially from near only where solid image brakes apart into separate fractals.

Fractal temple

In this raytraced image by unknown author we can see a strange sea temple. The temple consists of many similar platforms of various sizes. Every platform stands on four squared columns. The main cupola of the temple stands on platform which in one's turn stands on four platforms raised from water. There are four platforms smaller size on every of these main platforms with cupolas smaller size. Also, there are four platforms even smaller size on these small platforms and so on. This is a fractal temple with infinite count of cupolas!

Update 08.02.2009

Thanks to Greg M. Johnson, now I know who is the author of this image. His name is Christoph Hormann. The original image can be found at http://www.imagico.de/pov/gallery3_g.html.

Printgallery

M.C. Escher "Printgallery" (1956)

Printgallery is one of the most fascinating image of M.C. Escher. We see a young man which stands before print on the wall in print gallery. We see ships and seaside town with many houses on it. One house is closer to us than others and there's print gallery on the first floor. Then we look through the window of the print gallery and see young man looking at print. So we returned to beginning. It's a looped image, and we can infinitely look on it and never find the end.

But artist leaved center of the image empty and placed his sign there. What is under this blank spot? We don't know if Escher knew it. In 2002 this puzzle was solved and it's proved that reduced and rotated copy of the source image should be behind the spot.

Escher's "Curl-up" in paper

Tomohiro TACHI

This is origami version of M.C. Escher's lithograph "Curl-up" (1951). Author created four copies of a strange creature which turns into roller and sweeps away. In comparison, you can see original lithograph image at top right.

Metal sculpture by Kenneth Snelson

Kenneth Snelson creates his airy sculptures from metal. His structures are not hard joined but exist due to tension of individual elements. A term tensegrity was suggested structures of this type as combination of two words tension and integrity. Kenneth Snelson describes tensegrity as "a closed structural system composed of a set of three or more elongate compression struts within a network of tension tendons, the combined parts mutually supportive in such a way that the struts do not touch one another, but press outwardly against nodal points in the tension network to form a firm, triangulated, prestressed, tension and compression unit."

Cutting a wire in a tensegrity structure causes a major deformation or collapse.

Using principle of tensegrity he created many towers and other sculptures from aluminium and steel. You can see two photos of 60 feet high sculpture "Needle Tower" which was created in 1968. To the right is the front view of the tower and below is view from the basis of the tower.

Kenneth Snelson was born in Pendleton (Oregon, USA) in 1927. Educated in University of Oregon and Chicago Institute of Design. He had took part in many personal and grouped shows. His sculptures are in many collections and museums.

A reptile from Escher's artwork

A very interesting object was found at Google Maps near Ciudad Juárez in Mexico. It represents a single reptile from Escher's artwork "Smaller and smaller" (1956). Below you can see the original Escher's artwork and part of it with reptile in similar position.

This artwork is an artistic illustration of hyperbolic space. Due to properties of such kind of hyperbolic space you can infinitely go towards the center of the square but never reach it. Also Escher illustrated hyperbolic space in his artworks "Snakes", "Circle limit IV" etc.