Kaleidocycles, Part 6


The remaining hexagonal cycles. In the set, there are 17 separate papercrafts – the 6 geometric solids, 6 hexagonal Kaleidocycles, 4 square cycles, and the one twisted cycle.

The one on the right has a 2-color map, so it repeats after two turns. The one on the left has 4 unique faces per double wedge, so it repeats after 4 turns.

These are the last two hexagonals, showing all four turns. The one on the left has 4 different patterns, the one on the right has only 2.

Now listening to: Hawkwind — Warrior On The Edge Of Time


HAWKWIND — Warrior On The Edge Of Time (1975)

Kaleidocycles, Part 5



(All rights belong to their owners. Images used here for review purposes only.)

And now, the twisted cycle. This one is made up of 5 double wedges, and the folds are just off-center enough to introduce a self-occurring 180 degree twist. When you glue the ends together, it becomes obvious which way you need to turn the slot to get the full twist, because you’re going to have an optimist-optimist / pessimist-pessimist match up. In essence, you’re making a Mobius ring.

First tab glued.

Second tab.

Third tab.

Fourth tab.

Fifth tab.

The end of the finished fifth double wedge.

The cycle glued into a ring. There’s only the one twisted cycle in the box set. It’s no more difficult to construct than any of the other objects. It’s just takes longer, because of the extra fifth glue tab.

The finished cycle, after the glue cured. Visually, it doesn’t look as “clean” as the hexagonal and square cycles do, and it’s harder to see the animation as the ring is rotated through the center. To me, this is not a really successful experiment.


(With the ring rotated one panel.)

Now listening to: The Neutrons – Black Hole Star


THE NEUTRONS – Black Hole Star (1974)

Kaleidocycles, Part 4



(All rights belong to their owners. Image reprinted here from the book for review purposes only.)

I’ll get more into the construction of the Kaleidocycles here. We’ll start with the square cycle. The only difference from the hexagonal cycle is that you’re using 4 double wedges instead of 3. This means having 4 glue tabs, making for one extra step for gluing the paper together, and the finished ring has a square shape.

Folded sheet, with the first tab glued.

Second tab.

Third tab.

Fourth tab.

Ok, the next step is to make the ring. At one end of the cycle we have 2 folded, smaller glue tabs.

At the other end, the unglued edges of the final wedge form a slot. Apply glue to both sides of the smaller tabs and then slide them into the slot and wait for the glue to dry.

Actually, the book glosses over a major problem here. If you apply glue to both of the smaller tabs, when you make the ring one tab is going to be facing you and the other will be trapped inside the ring. Only the tab facing you will be exposed to the air for the glue to dry, and you have to constantly press the paper of the wedge against the tab to get the glue to stick. If you wait about ten minutes and then rotate the ring, the once-trapped panel will face you, and you’ll have wet glue smeared over everything. Plus, that tab adhered to the back of the first one, and there’s nothing for the trapped tab to glue against. Very messy and frustrating. It would have been nice if the authors had provide extra paper to the pattern to compensate for this. As it was, I always ended up with big seam lines from this last step.

The finished ring, with the last glue tab setting.

One day later, after the glue has fully cured. This is a simple 2-color map, so the cycle repeats after 2 turns.

Kaleidocycles, Part 3



(All rights belong to their owners. Image reprinted here from the book for review purposes only.)

Ok, now we get into the Kaleidocycles. The basic principle of a cycle is a double wedge. Take two identical pieces of paper cut into triangles, and tape them together along one edge. Take two more identical triangles and tape them together along one edge as well. Open up the two wedges, rotate one 90 degrees off from the other, and then tape the matching free edges together to make the double wedge. Repeat this process to make 3 or 4 double wedges, and tape their long edges together to make a chain. Finish by taping the opposite ends of the chain together to make a ring. The more interesting feature of this ring is that you can turn it through itself (rotate the corners of the wedges toward the center of the ring, or away from the center). As long as the hinge points (where the long edges of the wedges are taped together) are moderately flexible, the ring will rotate easily, with no damage to the paper.

The authors created 3 kinds of cycle – hexagonal, square and twisted. I’ll start here with hexagonal. The name refers to the shape of the assembled ring, and consists of 3 double wedges in total. There’s very little difference in difficultly making one cycle over the other, it’s just a question of how big the glue tabs are, and how many tabs you have to glue. If you look at the top photo, the paper is creased and ready to start gluing. The large white triangles are the glue tabs. Apply glue to one tab (I started from the left end) and fold the paper over width-wise to get the opposite panel to fit over the tab. Hold the paper in place until the glue dries, then go to the next tab. When all 3 tabs are glued, then glue the ends of the ring together (I’ll show that step next time.)

The finished hexagonal cycle. Each double wedge has 4 faces. If a tile has only two colors, then the cycle repeats after just 2 turns. The authors messed with the color maps occasionally to have 4 colors, giving a 4-turn cycle. On one cycle, they fudged by adding a fourth blank face with the words “M.C. Escher”, to get a 3-color mapped 3-turn cycle and then the filler text.

Now listening to: Yes – Tales from Topographic Oceans


Yes – Tales from Topographic Oceans

Kaleidocycles, Part 2


I’m going to spread these photos out over a few weeks. These things take about 1 hour each to make, largely because of the time it takes for the glue to dry. I’ve developed a kind of assembly line process, where I’ll make 2 or 3 objects at the same sitting, so that as the glue sets on the tabs of one object, I’ll use the time to fold or glue the other one. Initially, I was doing three of the geometric solids at one time, but when I got into the cycles, it was easier to just drop down to two. So, I’ll write them up as I work on them, and post the entries on Wednesdays, alternating with the “What’s playing” video links (youtube videos of obscure synthesizer albums).

These are the last of the solids – Icosahedron (bees), Dodecahedron (shells) and Cuboctahedron (fish). Again, there are problems with registration of the artwork over the cut and fold lines, so the finished papercrafts have white seams and misaligned lines. Also, as I was working with the paper, I’d be careful about making really strong creases, but as I glued the tabs together, the seams would soften and the finished objects would look rounder, more like balls. This was mainly an issue with the geometric solids having larger numbers of faces and that already look ball-like. Overall, though, it is fun making these and then looking at them.

The approach the authors used to tile the objects with Escher’s artwork is to identify points of rotation (where a specific object, like a bee or a bird) can be rotated around the beak or wingtip) and then draw straight lines through the rotation points across the full image. This will give you individual tiles, where one tile can be repeated over the entire surface of the object. This just leaves the choice of color map. Generally, Escher used 3, but sometimes 2 or 4 colors, the rule being that you can’t have two adjacent tiles of the same color. The authors followed this rule for the most part, but one or two of the Kaleidocycles required different mappings.

Looking at the above solids, the choice of tile patterns are pretty obvious.

Now listening to: The Pentateuch of the Cosmogony


The Pentateuch of the Cosmogony – (1979)

Kaleidocycles


(All rights belong to their owners. Images used here for review purposes only.)

Kaleidocycles, by Doris Schattschneider and Wallace Walker (1977, Box edition 2004)
I’ve long loved M.C. Escher’s works, but very few of the people I’ve talked to here in Japan seem to have heard of him. So I was surprised to see this book on the shelves at Junkudo. I ran home and did a net search to get a price check on it, and to look at the reviews on Amazon. One person complained that the authors had stamped their names all over the cards, but otherwise most of the other comments were positive. That, and the U.S. price was $14. The Japanese sale price was 1,200 yen ($13 USD including tax), so I went back and bought a copy the next day.


(Booklet cover.)

This is actually a large boxed coffee table book set. The box contains a 58-page booklet with examples of Escher’s most famous cyclic drawings, explanations of how the authors went about making the Kaleidocycles, and a detailed discussion of how the Kaleidocycles tie into geometry. Then, there’s a stack of printed, pre-cut, pre-creased sheets that you use to make the 6 different geometric solids and 11 Kaleidocycles.


(Back of booklet, with an example solid.)

A Kaleidocycle is a chain of polygonal objects that has one of Escher’s works crawling along the outside surface. They’re designed so that the creatures or patterns in the artwork are endlessly repeating as you rotate it through the center. The objects are all papercrafts that you punch out and glue together. This is really why I wanted to get this book – I want to make more papercrafts.


(Punched-out pieces for the Cube.)

I started with one of the easiest regular solids – the Cube. It has two punched pieces in one sheet of card stock. It’s a simple task of gluing the two pieces together to make a cube.


(The Cube, Tetrahedron and Octahedron.)

Because the Cube went so quickly (only ten minutes total) I checked the assembly list to see what else I could do in one sitting. I discovered that there was also a Tetrahedron and Octahedron. The Tetra was made up of only one sheet, but because of the way the glue tabs were laid out, it was kind of hard to reach inside to press down on the inner sides of the tabs to set up the glue. The Octa used two sheets and was somewhat easier to assemble because it didn’t want to close up until the very last side was put into place. But, what I didn’t like about the Octa is that it had registration problems. The punch out lines were offset from the artwork about an 1/8″, leaving a white strip along several of the edges. The fact that the white strips are in the middle of the dark red creatures makes them stand out even more. Although that’s the way the sheets were printed, it looks to the casual observer as if I made it wrong. Not happy about that…


(Same objects, just showing different faces. The key is to look at the corners to see how each of the creatures in that image have points of rotation, and at the edges to look for reflections.)

I’ll work my way through the other objects in ascending difficulty.
What I found interesting was that the bookstore employees had taped a photo of the finished objects to the front of the box so the customers could get a feel for what’s in the package, and there was a second small sheet taped on with a short biography of Escher in Japanese, explaining who he was. And, as for the one complaint on Amazon, the co-author’s initials are printed on the punch-out sheets in a relatively small corner or edge of the patterns to discourage people from scanning the artwork and printing out free copies. It’s something you kind of have to expect, and it isn’t that obtrusive into the artwork, for the most part..

Recommended if you like making papercrafts and you like Escher.

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