Paper Polygons and The Rosetted Logusz Cube
It's The Great Stellated Dodecahedron Charlie Brown! This is the first paper polyhedron I ever made. It also got me interested in mineral crystals and magnetic ferrofluid.
Make your own here: http://www.korthalsaltes.com/model.php?name_en=great%20stellated%20dodecahedron
A little bit boring, but necessary: Four of the Five Platonic Solids--and a hexaflexagon ring. Added my violin just to spruce up the picture. Platonic solids are 'regular' (same number of sides meating at each corner and the faces are all the same). There are only 5 of them (I made the 5th-an icosahedron-a while ago).
Linear dimensions are proportional to Linear dimensions;
Area is proportional to Linear dimension squared;
Volume is proportional to Linear dimension cubed;
and you can use this knowledge to vary the sizes of your models!
One of my favorite Rubik-style puzzle: the fluctuation angle. It starts add a cube then turns into jagged star shapes. I still haven't gotten it back to cube form--or one color per side.
This all led me to burr puzzles, named because they resemble plant seed burrs.
Angel Trumpet. We get around 40 of these huge spiked pods in the garden. You cut them in halfand they have tons of seeds in them. We plant them around my house and my parents house. The flowers are big. White Angel Trumpets. Plant and they come up the following year and TAKE OVER the flowerbed worse than pachysandras, lol. I'll never have to weed the flowerbeds again.
I made this a couple days ago: it's called a 'pentagrammic antiprism'. It looks really simple/boring, but paper cutting-wise it was probably one of the most complex shapes I've cut out, taping together was difficult.
Paper geometric shapes can actually be used, not just made and forgotten about. Johann Kepler’s nested polyhedra (this one de-nested with scissors for playing on a turntable/laser pointer) showing the the cycle of Jupiter & Saturn conjunctions (positional astronomy). Circling around, every empty space between each paper column equals almost 80 years, where the columns touch the top disk equals 4 conjunctions of the planets, so about every 20 years Saturn and Jupiter conjunct. Next on is in the year 2020.
I took at piece of paper and made this stage four Koch Tetrahedron (unpacked from a stage two). I placed it next to a Bismuth crystal (one my favorite-yet cheapest possessions). I have a theory that, although very rarely occuring in crystaline form Bismuth crystals inspired the design of Pre-Columbian step pyramids.
They used Bismuth in bronze weapon-making, and maybe on occasion they let some slag fall and cool slowly by the fire--producing these crazy irridescent crystals. Bismuth is a lot like lead (though not toxic-and even used in Pepto-Bismal). When you unpack a crystal (using math: via the Fibonacci number) or just by flipping every other level over (in paper models) you leave empty spaces: thus my two to four unpacking of the paper model--the Bismuth crystal is also hollowed out in nature (thus making it a Hopper crystal). The paper tetrahedron is about the size of my balled-up fist.
A Mayan pyramid? Iridescent man-made jewelry? Part of a circuit board? Nope! This is what happens when you heat the metallic element Bismuth until it is molten like hot Lead, and then let it slowly cool. Bismuth crystals can create stepped-pyramid shapes or boxes-within-boxes. Colors can range from mostly gold and silver chrome look, to iridescent oil-slick looking metallic blues, greens, purples and oranges.
The power and complexity of nature. Reminds me of the interior sets of the movie ‘Alien’ only more colorful. Bismuth is used as a substitute for the metal Lead in making toy soldiers, shotgun pellets and fishing lure weights. Bismuth is also used in Pepto-Bismol! Many large bismuth crystals are made German laboratories-but it is possible to make them on a household stove-top by melting and pouring between two pans as it cools.
Ancient South/Central Americans had Bismuth-and I think it may have influenced the designs of their stepped-pyramids; but then again what do I know? This crystal is big and heavy.
Here are some I just made on my stove. The bismuth I bought was too pure: impurities and oxidation age what cause the cool colors.
The power and complexity of nature. Reminds me of the interior sets of the movie ‘Alien’ only more colorful. Bismuth is used as a substitute for the metal Lead in making toy soldiers, shotgun pellets and fishing lure weights. Bismuth is also used in Pepto-Bismol! Many large bismuth crystals are made German laboratories-but it is possible to make them on a household stove-top by melting and pouring between two pans as it cools.
Ancient South/Central Americans had Bismuth-and I think it may have influenced the designs of their stepped-pyramids; but then again what do I know? This crystal is big and heavy.
Here are some I just made on my stove. The bismuth I bought was too pure: impurities and oxidation age what cause the cool colors.
All of this led up to me being interested in ferrofluid and creating my own polygon: The Rosetted Logusz Cube. Made a larger one by folding 12 isosceles triangle based oblique (non-right angled, irregular) pyramids and taping them together. The smaller one is a single piece of paper folded from a mathematical 'net'. Next will be some more permanent material.
While not maintaining their symmetry of Axes with a non-90° simple rotation-but a skewed one close to 45°, I also *inverted* some of the foldes to make the other cube disappear into the main one--instead of having the corners of both cubes poking out equally. One cube is subsumed on three corners by the other-but juts out in one corner--but also juts out in an inverted *bloom* as well: which is an inversion of a corner in the projection of an entire face of the internal (secondary?) cube. But each cube is still equal in volume~so we still have two equal cubes shown (the human eye is just drawn towards the 'main' one that still looks most 'normal'). I'm not sure-but I don't think this polyhedron has an actual name (many just have numerations of whatever they are anyway). So it's a "partially inverted-partially face tesselated compound of two cubes"...or 'The Logusz x2/45° +/- Face Rhombic' for short.
Also, it's a compound, nonconvex (and nonuniform) shape. So, not a polyhedron. Once it is stellated its lack of symmetry will be even more apparent. The solution? Interpenetrate 4 to 8 more Rosetted Logusz Cubes for symmetry! An octal-compound polyhedron!
While not maintaining their symmetry of Axes with a non-90° simple rotation-but a skewed one close to 45°, I also *inverted* some of the foldes to make the other cube disappear into the main one--instead of having the corners of both cubes poking out equally. One cube is subsumed on three corners by the other-but juts out in one corner--but also juts out in an inverted *bloom* as well: which is an inversion of a corner in the projection of an entire face of the internal (secondary?) cube. But each cube is still equal in volume~so we still have two equal cubes shown (the human eye is just drawn towards the 'main' one that still looks most 'normal'). I'm not sure-but I don't think this polyhedron has an actual name (many just have numerations of whatever they are anyway). So it's a "partially inverted-partially face tesselated compound of two cubes"...or 'The Logusz x2/45° +/- Face Rhombic' for short.
Also, it's a compound, nonconvex (and nonuniform) shape. So, not a polyhedron. Once it is stellated its lack of symmetry will be even more apparent. The solution? Interpenetrate 4 to 8 more Rosetted Logusz Cubes for symmetry! An octal-compound polyhedron!
Here's the polyhedral net for making your own:
R = Regular cube face.
O = Outward pink pyramids.
I = Inward pyramids.
A simpler way to think of this shape is to view it in its composite cell forms. Just like most stellated star-shaped polyhedra are really just an initial shape like a dodecahedron with added pyramid-shaped cells glued onto it, except that the Rosetted Logusz Cube is just a series of oblique pyramids! There is no actual center shape.
That's it! Just pyramids!
The base of each pyramid is an isosceles right triangle (90°, 45° and 45°).
The sides of each pyramid are right triangles (90°, x and y).
Here's a close-up of a triangle. If you made a bunch of pyramids out of wood the pink O pyramids and the R pyramids would just be solid pyramids. The I pyramids would be voids and actually the sides of the other pyramids, so: you would not make the I pyramids if making the Rosetted Logusz Cube out of actual solid pyramids.
The paper net needs the I pyramids because it's a hollow paper construction on a single piece of paper. You could make a bunch of paper pyramids and glue them together afterwards-you'll see you used less pyramids then the net version.
