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Thirty Interlocking Irregular Equatorially Truncated Triangularly Distorted Dipoles 270 units In my hand.
This is the next step building on the idea used in the 12-compound and moving towards 30 Irregular Cubes with 7 different unit types. (Obviously the proportions are going to have to be thinned dramatically to accommodate 90 more units.) In the mean time though, I do like the thickness of the proportions given the complexity of the design here- this reminds just a little of Aaron's 15 irregular hexahedra. As with the 12 compound, working with duplicated dipoles is harder than their corresponding polygons. I might consider a compound involving duplicated square dipoles in the future as well.
I know I haven't posted a lot recently, but this is not because I have not been folding- I actually have several projects ongoing at the present and have had this done for a week or two- I just haven't taken the time to upload a lot.
Designed by me.
Folded out of copy paper. (5 different paper proportions)
4 Interlocking Irregular Triangular Antiprisms 48 units 4-fold view.
There are a couple of compounds of 4 Irregular Octahedra, and this is one that has not been folded (as far as I know). It has been around a long time. The main challenge was getting a modestly clean assembly of 4 units per vertex where the narrowest angle is around 31 degrees. I named these triangular antiprisms because the octahedra are distorted along opposite faces.
Designed by me.
Folded out of copy paper.
“Obvolvi” 20 Intersecting Wrinkled Irregular Deconstructed Triangular Crystals 60 units
2-fold view.
This is much closer the project I originally had in mind a couple months ago, while initially resulted in “Kilauea.” At that point, I noted that the original compound “remains to be folded.” Other than the chirality of the outermost 5-fold axis whorls, this is exactly what I initially had in mind: a woven solid where wrinkled edges which begin in triangular “3-point” intersections representing a vertex of a dodecahedron wrap around the dodecahedron to terminate at the three vertices which are each adjacent to the vertex which is the polar opposite of the starting vertex. The assembly here was very enjoyable, and was a little trickier to figure out than I initially anticipated. It would be nice to give this a proper 5-color sequence, but I’m not sure that I will go to the effort. The geometric name is again contrived- this is a woven solid, but one way of visualizing it is as a compound of 20 triangular “crystals” where one vertex of each “crystal” is deconstructed so that the edges intersect other vertices. The units wrap around the surface of a dodecahedron, hence the nickname: “obvolvi” can mean “to be wrapped” in Latin. There is a cheatsheet if anyone wants one.
Designed by me.
Folded out of copy paper.
Thirty Interlocking Irregular Hyperboloidal Polarly Diminished Quasi-Crossed Triangular Prisms 210 units 5-fold view.
I had mentioned back in November that I had not ruled out a 30 compound of the frames used in this model: www.flickr.com/photos/byriahloper/38590768552/in/datepost...
That being said, when I began this project, I had not planned on it being one of the most visually complex compounds every created.
Each polyhedral frame here is a rather unusual shape composed of 7 units. Essentially, the frames are a tetrahedron where two adjacent triangular faces are rotated in opposite directions along an axis running through the one edge that is not part of either triangular face. Then, reflectively symmetric vertices are joined, essentially making an "x" shape where the original edge was that both adjacent triangles shared. Sorry if this doesn't make a lot of sense- the transformation isn't actually as complicated as it sounds.
So why did I name them as a triangular prism distortion? It is probably most convenient to view each frame as a triangular prism with 1 side crossed, and 1 edge removed from each "triangular face".
That aside, the model itself is pretty much as difficult as it appears. It could still use a few alternations (this is version 2.75), so I won't do a cheatsheet yet, but I plan to do one eventually. This has 4 different paper proportions.
Designed by me.
Folded out of copy paper.
Thirty Interlocking Irregular Hyperboloidal Polarly Diminished Quasi-Crossed Triangular Prisms 210 units 2-fold view.
I had mentioned back in November that I had not ruled out a 30 compound of the frames used in this model: www.flickr.com/photos/byriahloper/38590768552/in/datepost...
That being said, when I began this project, I had not planned on it being one of the most visually complex compounds every created.
Each polyhedral frame here is a rather unusual shape composed of 7 units. Essentially, the frames are a tetrahedron where two adjacent triangular faces are rotated in opposite directions along an axis running through the one edge that is not part of either triangular face. Then, reflectively symmetric vertices are joined, essentially making an "x" shape where the original edge was that both adjacent triangles shared. Sorry if this doesn't make a lot of sense- the transformation isn't actually as complicated as it sounds.
So why did I name them as a triangular prism distortion? It is probably most convenient to view each frame as a triangular prism with 1 side crossed, and 1 edge removed from each "triangular face".
That aside, the model itself is pretty much as difficult as it appears. It could still use a few alternations (this is version 2.75), so I won't do a cheatsheet yet, but I plan to do one eventually. This has 4 different paper proportions.
Designed by me.
Folded out of copy paper.
“Soccer Ball” Six Interlocking Pentagonal Prisms + 10 Interlocking Irregular Hexagonal Prisms 270 units
3-fold view.
This is a model that I have wanted to design for many years- almost as long as I have been designing wireframes, but I have until the past year lacked both the ability to comprehend how the interior weaving might work, and also the physical tools to implement the design. Neither of those are nearly as much of a problem anymore.
This is a logical extension of prismatic series that Daniel Kwan began so many years ago. A quick look at the 3-fold orthogonal projections of a truncated icosahedron vs a regular icosahedron and dodecahedron (upon which Daniel’s first compounds of 10 triangular prisms and 6 pentagonal prisms are respectively based) will show you exactly why he stopped where he did with prisms. Opposite hexagonal faces are not rotated as an icosahedron’s triangles or a dodecahedron’s pentagons are. Without that rotation, weaving is much more challenging. What to do?
You can force the rotation into the frames themselves so that opposite faces are rotated. I did this in 6 Rhombic Prisms, then modeled the 4 point intersections. (You have to model the intersections because the distortion of twisting a prism works great on the ends of the prism, but as you get towards the middle, the rotation diminishes to nothing.) Or, you can force rotation with wrinkles, as I did in 15 Rhombic Prisms. Crude but effective.
Here, you have hexagonal prisms, which can be “divided” into 2 sets of points where each set represents the vertices of a triangular prism. We already know 10 triangular prisms weaves, so as long as the 2 “triangular prisms” are not the same size (because if they are the same size then they will occupy the same space within the construction, because the midpoint of each edge will be the same distance from the central origin), the compound is weavable. If the “triangular prisms” are not the same size, the consequence with the actual hexagonal prisms is that the faces are irregular because every other vertex is a different distance from the vector which defines the axis of orientation for the prism (ie. the very center of the hexagonal face).
The closer the hexagonal prism is to a triangular prism, the more comfortable the center edges sit, but then you lose the truncated icosahedron in the bargain. Therefore, the goal here was to make the hexagonal prisms as regular as possible to enhance the affect. To do that, I made the prism edges about as thin as I possibly could.
I describe a wireframe compound with more than one type of frame as a “Composition” (not to be confused with composition as the ordering of Euler rotations in 3D), and these are frequently somewhat tacky in my opinion, but I am beginning to see that there may actually be cases in which the use of multiple frame types complement each other. I used variable width edges here to highlight the truncated icosahedral nature, but I also decided to not make them solid so as to obscure the weaving within. I think the compromise was a pretty good one. An octahedral version of this should be possible as well.
Designed by me.
Folded out of copy paper.
6 Interlocking Irregular Hyperboloidal Gyrobifastigia 84 units
2-fold view. It was long past time that someone do a compound with these wonderfully simple Johnson solids. The center edges of each gyrobifastigium do form a standard compound of “6 squares”, but since the structure has tetrahedral symmetry, the “squares” are actually distorted to rhombi. Thus assembling this involves a bit more than simply tacking on half of a triangular prism to each side of a compound of 6 squares. I’m sure there are other compounds with wider proportions, but I was specifically looking for a compound that was not “too tetrahedral” in shape, and had some interesting weaving interactions (which is not so easy to do these days with only 84 units to work with). These days, I do consciously try to incorporate “pseudo-axes”, though I have not had the time (or likely the skill) to develop an algorithm for their specific inclusion in edge interactions. I can easily imagine other compounds of 3, 12, 15, 24, and 30 gyrobifastigia, but the central four unit intersections are a real killer- compounds relying on central stability would not lend themselves easily to gyrobifastigia, as the locks at such intersections are either weak or difficult to assemble. This model could use a refold with slightly thinner proportions, which I might do at some point in the near future.
Designed by me. Folded out of copy paper. (8 different proportions)
Six Interlocking Irregular Monotruncated Octahedra 96 units
In my hand.
My first new compound since the convention, I have thought about the general idea of of a 6 octahedra compound for some time, and last year tried a 6 square pyramids compound, where rather than being a trivial variation, the pyramidal points went inward into the center of the model. It didn't initially work, and rather than continue, I gave up on it. However, a few weeks ago, I realized that the details could be easier worked out on POV-ray if I rendered it as 3 interlocked sets of 2 opposite frames each. The "monotruncated" part was not in the original plan, but the units joining at the outer vertices would have rather narrow angles. Rather than battle through points that would be painfully difficult, and very visible on the model, I added some little truncations. Despite putting an added 24 units on to the total, these were definitely the way to go. The truncations made the frames look a lot like lanterns, hence the nickname.
Designed by me.
Folded out of Cordenons' Stardream paper.
For two months couldn't find a moment to take a picture of this unusually shaped apple. It was surprisingly keeping its shape in spite of wrinkles.
First image of 2015 after the holiday bustle was over.
Twelve Interlocking Irregular Triangular Prisms v.1 108 units 4-fold view.
I have been extremely busy with my classes, and thus have been working on this design slowly in the tiny break spaces I have tried to carve out.
This design is actually fairly simple, despite having 5 different unit types. Basically, one edge on one side of a triangular prism here represents one edge of an octahedron (or alternately, one rectangular face of each prism represents one rectangular face of a cantellated octahedron). In fact it is quite similar in design style to the "30 Triangles" and "42 Pentagons" compounds I have made in the last year.
This particular fold is the first version, and is just a little bit on the tight side, so I will probably refold this at some point soon.
Designed by me.
Folded out of copy paper.
“Radioactive” Ten Interlocking Irregular Hyperboloidal Triangularly Distorted Nonagonal Prisms v.3 270 units
5-fold view. Since the completion of 10 Trapezo-rhombic Dodecahedra in October, this has been my primary Wireframe project. Last year I experimented with “Shaft-twist” edge units, in which the orientation of the unit’s dihedral angle is effectively rotated 180 degrees. Of course, when working with star polyhedra, the angle becomes wider at the base of the points as the number of total points on the prismatic faces are reduced. 5 points was very reasonable, but 4 points was challenging, and 3 points (which would make distorted hexagonal prisms, was seemingly impossible. However, by truncating the points, the narrow angles of 3 point stars were obviated, thereby making an irregular “3 lobed “ nonagonal prism whose faces each look somewhat like a “radioactive” symbol, hence the nickname. This pattern would obviously make possible a 2 lobed prismatic construction as well, whose faces would have an “hourglass-like” shape.
This might be one of the most challenging models I have designed in the last few years. The weaving itself is fairly simple (or at least straightforward), but the units are complex and time consuming, and the assembly process is outstandingly difficult. I had to redesign the locking mechanism for the inner units, since the first assembly would not hold up. The new locks are much stronger, but consequently are more difficult to assemble. Given the number of inner vertices and their placement, the inner units must be assembled first, but then they have nothing to support them. I used a scaffolding dodecahedron, but if the scaffold is not perfectly sized, the edges will be in the wrong place. This will then negatively affect the later assembly.
I rendered the compound in POV-ray, and calculated all of the proportions and interior angles out in advance, but calculations based on a render could only get me so far in this case. The final model turned out modestly well, but I don’t think I will be refolding it to iron out the details any time soon.
This model uses 5 different paper porportions.
Designed by me. Folded out of copy paper.
me as Miyuki Shiba from The Irregular at Magic High School (Idol Version).
photo by Butterfly Dreams (facebook.com/ButterflyDreamsCosplay)
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DO NOT USE WITHOUT PERMISSION!
Sharing just with FULL credit of cosplayer's and photographer's name and website link !!!!
Thirty Interlocking Irregular Equatorially Truncated Triangularly Distorted Dipoles 270 units 2-fold view.
This is the next step building on the idea used in the 12-compound and moving towards 30 Irregular Cubes with 7 different unit types. (Obviously the proportions are going to have to be thinned dramatically to accommodate 90 more units.) In the mean time though, I do like the thickness of the proportions given the complexity of the design here- this reminds just a little of Aaron's 15 irregular hexahedra. As with the 12 compound, working with duplicated dipoles is harder than their corresponding polygons. I might consider a compound involving duplicated square dipoles in the future as well.
I know I haven't posted a lot recently, but this is not because I have not been folding- I actually have several projects ongoing at the present and have had this done for a week or two- I just haven't taken the time to upload a lot.
Designed by me.
Folded out of copy paper. (5 different paper proportions)
NGC 4449 is a relatively nearby galaxy located in the constellation Canes Venatici. It is classified as an irregular galaxy, as is the Large Magellanic Cloud, a satellite of the Milky Way. Both have distinctive bar-like shapes in their cores.
In this image from Hubble’s Advanced Camera for Surveys, hot bluish-white clusters of massive stars are seen scattered throughout the galaxy, interspersed with numerous dustier, reddish regions of current star formation. Massive dark clouds of gas and dust are silhouetted against the starlight.
NGC 4449 is experiencing unusually intense star formation. Such "starbursts" usually occur in the central regions of galaxies, but NGC 4449 has more widespread star formation activity, since the youngest stars are observed both in the nucleus and in streams surrounding the galaxy.
A "global" starburst like NGC 4449 resembles much older, "primordial" star-forming galaxies, which grew by merging with and accreting smaller stellar systems. It's likely that the current widespread starburst in NGC 4449 was triggered by interaction or merging with a smaller companion.
For more information, visit: hubblesite.org/image/2155/news_release/2007-26
Credit: NASA, ESA, A. Aloisi (STScI/ESA), and the Hubble Heritage (STScI/AURA)-ESA/Hubble Collaboration
Four Interlocking Irregular Ditrigonal Triangularly Distorted Hexgrammically Paraugmented Hexagonal Prisms 72 units In my hand.
I originally sketched out a series of cyclically symmetric hybrid self-interlocking prism/star prism compounds years ago, and recently decided to wander off from my main projects for a few days in order to pursue the simplest of these ideas. Essentially, what this resulted in is “Gasherbrum” + “4 Interlocking Triangles #2” + “4 Interlocking Hexagons” all connected together with the edges of a hexagonal prism. Of course, octahedral, icosahedral, and dodecahedral versions should be possible as well, which I may get around to at some point in the future. Describing these shapes geometrically is a bit awkward: “ditrigonal” refers to the fact that the edges of the hexagonal prism are grouped into 2 sets of 3 edges, “triangularly distorted refers to the fact that 3 of the 6 edges are lower and wider (or higher and thinner, depending on how you want to look at it) than the other 3, and “hexagrammically paraugemented” refers to the fact that one side of each prism is a hexagonal face while the other is a hexagram (or would be, if it were not distorted). The prefix “para” thus refers to “along the side of” which is one face of each prism. Apologies if that does not seem very clear- as anyone can see, this is not that complicated an idea, it is just difficult to describe geometrically.
Designed by me. (5 different paper proportions)
Folded out of copy paper.
Three Interlocking Irregular Crossed Square Antiprisms 96 units 2-fold view.
A week or two ago I sketched out a pattern for a 3D shape implementing the polar cyclical symmetry of the “thingies” compounds from years ago, and after a little work, this is the result. I must say that I am surprised that no one has done any work in the area of designing wireframe compounds based on the topological distortions used to make crossed antiprisms.
Anyway, this was fun to fold, and even though it is not quite as finalized as I would like, I may try a comparable pentagonal compound (I have already rendered a dodecahedral compound on POV-ray.)
Designed by me.
Folded out of copy paper.
5 Interlocking Irregular Rhombicuboctahedra (v. 3.5) 240 units 2-fold view.
I have long intended to march through 5-compounds of the rest of the viable Archimedeans, and since “Stella Media” necessitated a POV-ray rendering of a truncated icosahedron, I thought I might repurpose the render for a 5-compound. However, this compound of rhombicuboctahedra was rendered long ago and I figured it was well past time that I rendered it first as a warm up. I first conjectured about this compound 12 or 13 years ago, and started a version of it about a decade ago based on a rendering Daniel Kwan made. However, it proved challenging and so was moved to the back burner where it stayed until the beginning of this year. When I started it afresh, it again proved a rather belligerent compound to render, and I had to tweak the original design and modify the original proportions considerably. When the first version failed, I repurposed the units for “Aldebaran.” In the next version, I thinned the proportions, simplified the pocket angles and proportions, and adjusted the pocket angles to narrow the dihedrals on all the rectangular faces, but the inner triangles were still too tight, so I disassembled everything yet again, and adjusted the pocket angles of the interior triangles to decrease the dihedrals there as well. By this point, the paper is fairly tortured and the model is still very rigid, but overall, I think this result is fairly decent. Each rhombicuboctahedron is, of course, tetrahedrally distorted. Further analogous compounds await.
Three different paper proportions.
Designed by me. Folded out of copy paper.
El tiempo estaba super loco cuando hice estas fotos. Cuando vino esta unidad diluviaba, y cuando vino el doble salio el sol. En la foto se ve a la 447 que pille en una anterior foto con un cercanias a Irun. Hasta arriba de viajeros debido a los Kilometroak 2013.
Ten Interlocking Irregular Hyperboloidal Triaugmented Omnitruncated Digonal Dihedra 150 units
In my hand.
This, is, undoubtedly, the most time consuming project I have ever designed. I folded it three times in total to get the perfect proportions and angles, and combined with the design process, this took almost 40 hours to perfect. The compound is unique- it was a compound of ten irregular hexahedra before, but with the addition of thirty units which connect the frames, it has become something special. Tracing the connection between the frames is quite entertaining. I can only imagine at future presentations the number of people who will be scratching their heads trying to figure out what exactly is going on here. I'm not entirely sure myself. xD The exterior vertices roughly represent the vertices of a stellated rhombic triacontahedron...hmm.
Designed by me.
Folded out of Zanders' Elephant Hide paper.
Three Interlocking Irregular Crossed Square Antiprisms 96 units 4-fold view.
A week or two ago I sketched out a pattern for a 3D shape implementing the polar cyclical symmetry of the “thingies” compounds from years ago, and after a little work, this is the result. I must say that I am surprised that no one has done any work in the area of designing wireframe compounds based on the topological distortions used to make crossed antiprisms.
Anyway, this was fun to fold, and even though it is not quite as finalized as I would like, I may try a comparable pentagonal compound (I have already rendered a dodecahedral compound on POV-ray.)
Designed by me.
Folded out of copy paper.
Four Interlocking Irregular Hyperboloidal Triaugmented Septahedra 72 units 3-fold vertex view.
This design uses the technique I have employed with triangles a number of times during the past on only the triangular faces of a truncated tetrahedron. I could have made this a compound of 4 truncated tetrahedra, but really, what fun is that when the edges can be extended and inverted to make a shape that is in some sense self- intersecting. There is more I would like to do with this compound, which I might look into in the future.
This has 6 different types of units, and 3 different proportions.
Designed by me.
Folded out of copy paper.