Q-BA-MAZE

Fish Sculpture at the Walker Art Center

A group of students, architects and sculptors got together recently to plan this Q-BA-MAZE installation at the Walker Art Center in Minneapolis. Over the course of an intense weekend of activity, the vague idea of making a fish resulted in this giant fish at the head of a growing school of smaller Q-BA-MAZE fish. (Construction plans for some of these fish are available on the plans page of the Q-BA-MAZE website).

Here is a night shot of the fish. It is tucked under this cantilevered ceiling of a recent addition to the Walker by architects Herzog and DeMeuron. (Notice the size of the fish compared to the guy doing calisthenics outside!)

Matt Peiken created this video blog for the Walker.

We had 10,000 cubes on hand for the installation and around 5,000 ended up in the big fish. Each of the vertical stripes of color in the fish is a separate section. This division made it easier to break the project into separate tasks. On our main installation evening, seven people arrived and worked on the fish simultaneously.

The curve of the fish is made by having each section slightly rotated with respect to its neighboring sections. The tail and fins are all just a single cube wide and held together just with the Q-BA-MAZE joinery. There is no glue in this construction. It is composed solely of the three types of Q-BA-MAZE cubes.

Design Thinking with Cubes, Components, and Constructions

The photo above shows the design called Q50/plan01 disassembled into several components. The components are made from two or more cubes, and these components can be rearranged into new constructions.

If you take a close look at the plan that comes with each 50-pack or at this plan from the Q-BA-MAZE website, you can see which cubes make each of the components shown in the top photo:

• Cubes 1-7 are the "base" (front middle in the photo)*
• Cubes 8-11 are a "4-cube scrambler" (back row, far right in photo)
• Cubes 12-22 are a "9-cube scrambler" with a "2-cube switch back" on top (back row, second from right in photo)
• Cubes 23-34 are a "double helix" (back row, third from right in photo)
• Cubes 35-36 are a "2-cube column" (back row, far left in photo)

* This "base" composed of cubes 1-7 is good stable starting point for any new construction.

The photo below shows Q50/plan01 on the left and a completely new construction on the right made from the same components just rearranged in a new order. Working with components like this is a faster way to design and build new constructions compared with building one cube at a time. An understanding of components will also assist your design thinking as you imagine your own new constructions.

Have fun inventing and building with Q-BA-MAZE!

Cantilevers, Counter-balance, Components and other Craziness

This photo shows a blue "straight-away" component cantilevering to the left, while a green "straight-away" component counter-balances the structure by cantilevering back to the right.

In this post, I illustrate a number of different "components" that can be built into a complete construction. I use the base configuration from Q20/plan03 (cubes 1-6) and then 4 blue single-exit cubes and 4 green single-exit cubes to create the various components.

This is a six-cube "zig-zag" component. Each green cube makes a right-turn and each blue cube makes a left-turn. The left-right-left pattern makes the zig-zag. An eight-cube "zig-zag" would have required further cubes on the opposite side as a counter-balance.

This is an eight-cube "switch-back" component. Each single exit-cube aims the pathway back under the previous single-exit cube. The result is very stable tower form that is just one cube wide and two-cubes deep.

This is an eight-cube "switch-back helix" component. The blue single-exit cubes each rotate 90 degrees with respect to the blue cube above. The green single-exit cubes aim the pathway back under the previous blue single-exit cube. The result is what appears to be a central blue column wrapped with a sporadic green helix.

This is an eight-cube "1x3 switch-back" component. The component is one cube wide and three cubes deep. The pathway goes straight through two cubes and then the third cube bends the path back under the previous two cubes. The set of four green cubes could be rotated 90 degrees with respect to the blue cubes and then you'd have a "rotating 1x3 switch-back".

This is a six-cube "straight-away" component. The two cubes on the left are necessary to counter-balance the weight of the "straight-away" as it leans far out to the right. It is not necessary for a structure to be symmetrical to have balance. There just needs to be enough weight to keep things from tipping. As you get more daring with your own designs, you'll have to experiment with trial and error. If your structure falls down, you probably went beyond some physical limit. Pushing these limits is part of the fun!

This is an eight-cube "helix" component. More specifically, this is a "2x2 counter-clockwise single-helix." Each cube turns the pathway to the left as it goes down, so the pathway spins counter-clockwise. If each cube turned the path to the right, it would be a clockwise helix. Looking at this helix from above, it fits on a 2x2 cube grid.

This is an eight-cube "2x2 counter-clockwise double-helix" component. It is much like the single-helix, but a second helix fills in the empty cantilevered spaces of the first helix. Here, one helix is green and the other is blue. DNA is a double-helix. Question: Is the DNA double-helix clockwise or counter-clockwise? and why?

This is an eight-cube "3x3 counter-clockwise double-helix." The two helixes will never touch. Double-exit cubes can be used occasionally as a "3x3 double-helix" is built as a means of connecting and stabilizing the two pathways. Cube #34 in Q50/plan01 is a double-exit cube used in this way.

This is an eight-cube "3x3 counter-clockwise single-helix."

The list of components goes on and on. Take a look at the post on 10 Billion Trillion Combinations and you will get an idea of just how many configurations are possible.

Have fun experimenting and exploring and finding the coolest components for your crazy constructions!

10 Billion Trillion Combinations: Toy Math and Design Space

WHAT IS A DESIGN SPACE?

The huge sets of possible permutations for LEGO Bricks and Q-BA-MAZE cubes are called "design spaces" (Beinhocker, p. 193). It is up to the designer, the person playing with LEGO or Q-BA-MAZE, to discover the best designs among the zillions of possiblities. The enormity of these "design spaces" describes both the potential challenge and the level of freedom for the designer.

How many combinations are there? How big are these design spaces? Just six 2x4 studded LEGO bricks of a single color can be rearranged in 102,981,500 different configurations (Bedford, p. 19). The Rubik's Cube* can be scrambled in 43,252,003,274,489,856,000 ways (Walsh, p. 230). The LEGO "Creator" set contains 500 pieces of different shape and color which can be combined in roughly 10 to the power 120 combinations (1 followed by 120 zeroes) (Beinhocker, p 193). Given that the universe contains around 10 to the power 80 atoms (1 followed by 80 zeroes), the 500-piece LEGO set is pretty impressive!

I've always been impressed with these huge numbers, but also a little skeptical, because I like seeing proof. It seems the math behind these figures is never shown. And probably, the math cannot be shown, because the problems are so complex that a computer program needs to be written to calculate the combinations. This is the case with Q-BA-MAZE. I can't easily get the answer to the seemingly simple question, "how many ways can the Q-BA-MAZE cubes be reconfigured?" because it would take a custom computer program to calculate this.

It is possible, however, to make a rule that describes a subset of the ways Q-BA-MAZE cubes can be rearranged and to express this rule in a simple mathematical formula. The results of our calculations, just a subset of the total possibilities, yield these surprisingly huge numbers of ways 18 to 36 Q-BA-MAZE cubes can be rearranged:

Calculation 1B above of the 18 single-exit cubes in a 50-pack yields 9,656,357,112,229,430,000,000 and can be described as roughly 10 billion trillion combinations.

The helical construction of Q-BA-MAZE cubes in the photo at the top of this post is a single continuous pathway with no jumps. It is made with the single-exit cubes that come in the Cool Colors 50-pack (Q50C) and is one of the configurations included in both Calculation 1A and 1B.

HERE'S THE MATH

To satisfy my curiosity, and with the help of some mathematician friends, we devised this formula:

If you would like to see this formula in action just open up this Excel file and you can manipulate it as you like and see what happens as you change the variables. Here is a screen shot of the Excel calculations:

Five pathways may converge on every Q-BA-MAZE cube in a construction. This formula, however, describes only a single pathway entering any particular cube, or even just a stacked connection. So the results of this formula are far lower than what an eventual computer program will find, but it will at least provide a minimum starting point for understanding how many ways the cubes may be reconfigured.

• N is the total number of cubes in a construction
• Nb = number of blue cubes, Ng = number of green cubes, Nc = number of clear cubes
• N = Nb + Ng + Nc
• C = number of connections
• when C = 4, there is a single continuous pathway with no jumps (the side joint on each cube is always engaged with the cube below)
• when C = 8, the path may be discontinuous and jumps are allowed (either the side joint or the bottom pegs may be used to connect to the cube below)
• the ! symbol means 'factorial' (as an example, 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720)
• the left side of this equation deals with the color combinations
• the right side of this equation deals with the cube configurations

* I include Rubik's Cube here because I found this huge number that describes it and so it makes a good example of how many ways something can be scrambled. Because Rubik's Cube is a puzzle, it is mostly thought of as having only one solution: all sides a single color. But there are interesting checkerboard and other patterns in the design space of 43 quintillion Rubik's combinations, its just that the mechanism of rotating cubes intentionally makes these difficult to find.

If you have an answer to the question "How many ways can the cubes of the 50-pack be reconfigured?" I'd be very interested in hearing from you!

Sources for this post:

The Unofficial LEGO Builder's Guide, Allan Bedford, 2005, No Starch Press Inc., ISBN 1-59372-054-2

The Playmakers: Amazing Origins of Timeless Toys, Tim Walsh, 2004, Keys Publishing, ISBN 0-9646973-4-3

The Origin of Wealth: Evolution, Complexity, and the Radical Remaking of Economics, Eric D. Beinhocker, Harvard Business School Press, ISBN 13-978-1-57851-777-0

Desktop Sculpture in Stone or Polycarbonate

photo: cover of the book Worlds Within Worlds: The Richard Rosenblum Collection of Chinese Scholars' Rocks

DESKTOP SCULPTURE

Q-BA-MAZE is a new form of desktop sculpture, but the idea of desktop sculpture for contemplation goes back more than 1000 years in China -- in the form of 'Scholar's Rocks'. Chinese scholars collected 'fantastic' rocks and displayed them on wooden bases. The rocks might look like clouds, landscapes, people, leaping tigers and so forth, or they could just be abstract forms.

The World of Scholars' Rocks exhibit in 2000 at the Metropolitan Museum of Art (with about 30 stones from the Rosenblum Collection) was my formal introduction to the subject. I say 'formal', because several years earlier, I had unwittingly purchased a scholar's rock of my own while rummaging in an alleyway antique shop in Guangzhou, China (while on a weekend break from my job in the Architecture Department of the Chinese University of Hong Kong). Having collected rocks since childhood, this richly veined stone struck me as the perfect souvenir. Only with the exhibit did I come to realize that my stone from Guangzhou was likely a scholar's stone separated from its base.

In the introductory essay of, Worlds Within Worlds, Robert Mowry writes, "Chinese scholars' rocks might be characterized as favored stones that the Chinese literati and their followers displayed and appreciated indoors, in the rarefied atmosphere of their studios." (p 19). They are displayed "indoors on desk, table, or bookshelf, though an especially large example would be set directly on the floor. Regarded as 'stand-alone' items, scholars' rocks are shown individually and are characteristically presented on carved wooden stands -- like fine bronzes and porcelains -- in order to orient and support the rocks and to distinguish them from the mundane." (p. 20).

The stone on the Worlds Within Worlds cover (top) is a stone from Lake Tai (Taihu), near Suzhou, China. The stones may either be found in nature or carved to appear natural -- Richard Rosenblum considered this particular example a carved stone. Stones with good holes all the way through are particularly prized. Mowry writes, "Although the lake (Taihu) produced limestone rocks with naturally dissolved holes and sand-washed surfaces in early times, the supply of such rocks had been considerably depleted by the late Tang period. By the Northern Song, local families had begun to sculpt rocks from the abundant native limestone, after which they placed their creations in the lake for several decades of natural finishing." (p.27)

THOUGHTS ON COMPOSITION

This Taihu stone is a dynamic form -- it appears to be moving like rising steam, a blowing cloud, or vaguely like a human arm bent at the elbow. It cantilevers boldly to the left and the weight of this gives the sense that it could overturn (counterclockwise). The red line at 'B' highlights the angle at which the wooden base meets the stone. This meeting line is rotated clockwise from horizontal in a way that visually balances the counterclockwise overturning implied by the cantilever. In opposition to these dynamic elements, the underside of the cantilever is horizontal (as demonstrated by line 'A'). If this underside were tipped one way or the other it would imply a movement. But the horizonality suggests stability and rest -- like the horizontal line made by still water in a glass.

Notice what happens if this Taihu stone is slightly adjusted, so that the wooden base meets the stone with a more horizontal connection. This forces the bottom of the cantilever to hang at an angle. It makes it appear as if the entire structure is wilting, drooping under its own weight, as opposed to the original compostition which appears to be floating upward like smoke.

Rosenblum also makes this interesting final point in his essay, "I am most fascinated by the thought that unlike other art, scholars' rocks are not fixed objects. They can be, and often have been remounted. They can be set in different positions, conjuring up new images that change their character altogether -- in effect, remaking them. This was something the Chinese did, and I have begun to do as well. Interacting with the rocks, moving them about, creating mounting environments for them and out of them, has brought me back to sculpture." (p 121)

The bottom image is a Photoshop simulation of the Taihu Scholar Stone remounted and utterly transformed.

And finally, a translation of the original composition into Q-BA-MAZE cubes:

 

The White Stripes: Living the Lego Dream

This is probably the coolest Lego thing ever. Lego forming a swimming pool with colorful spiral waves! A lead guitarist made from just 34 bricks and animated!

It is the video for "Fell in Love with a Girl", which is the second single released from The White Stripes' third album White Blood Cells. Released in 2002, it reached number one.

I have a DVD with this and other videos by the director Michel Gondry. It comes with a little book titled:

I've been twelve forever

The title is Gondry's self description. He is now 44 years old and the director of the films Eternal Sunshine of the Spotless Mind (2004) and Science of Sleep (2006).

Isn't it funny that we grow up with adult voices telling us to "stop acting like a twelve-year-old" only to reach adulthood and to find one of the greatest industrial designers of the 20th Century stressing the importance to the design process of "the attitude of the child"? (See my Sept 1 post). And stepping into the present, to find Gondry producing such brilliant work by having just remained twelve (at heart)?

Here is the "making of video":

*the next coolest Lego thing I've seen is a Lego robot that can solve the Rubik's cube (I'll save that for another post)

Angels and Firecrackers

Charles Eames once said that in the "world of toys he saw an ideal attitude for approaching the problems of design, because the world of the child lacks self-consciousness and embarassment."

When I came across this statement in The Work of Charles and Ray Eames: A Legacy of Invention (p. 139) it really jumped out at me. I have been doing a lot of play-testing with kids and adults and I have noticed how much more quickly kids learn. Children just proceed and experiment, they figure things out as they go along, they don't worry about rules, judgement or success.

The Eames House of Cards

The "world of the child" comment, led me to notice an underlying connection between the Eameses architectural design and their toy design (see this link for lots of photos of the interior and exterior of the Eames House). Both the Eames House and their toy the House of Cards have simple repetitive structural systems. The structural systems work, but it is the play of color and the collections/images of diverse things (in the house and on the cards) that brings them richness and meaning. Playing with the cards and living in the house are similar activities -- both involve a continual rearrangement of things, a richness of ideas that can come together in ways which inspire new unexpected and creative thoughts. Look closely at the photo here. Who ever thought of "angels and firecrackers in an archway"? These things don't go together. Such a combination is against the rules, but there are no rules in "the world of the child."

For more information on Charles and Ray Eames, see this website related to the Legacy of Invention exhibition organized by the Library of Congress and the Vitra Design Museum.

Cube Toy Mash-Up!

Kubricks and Q-BA-MAZE mashed together in an unexpected meeting! This is a cell phone photo of a set of Reservoir Dogs Kubricks displayed on a Q-BA-MAZE construction. Sam of Minneapolis, a long-time toy collector and a fan of the ROBOT Love design and toy store, just sent this in. It looks like some steel balls are flying by -- great action shot Sam!