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The Deconstruction of the Compound of Five Cubes
To make the compound of five cubes, begin with a dodecahedron, as seen above. Next, add segments as new edges, and let them be all of the diagonals of all the dodecahedron’s faces. Then, remove the pentagonal faces, as well as the original edges. What’s left is five cubes, in this arrangement.
Using polyhedral manipulation software called Stella 4d (available at www.software3d.com/Stella.php), these five cubes can be removed one at a time. The first removal has this result:
That left four cubes, so the next removal leaves three:
And then only two:
And, finally, only one remains:
Because their edges were pentagon-diagonals for the original dodecahedron, each of these cubes has an edge length equal to the Golden Ratio, (1 + √5)/2, times the edge length of that dodecahedron.
An Alphabetical Listing of Known Exotic Atoms
- Antiprotonic helium: an atom of helium, with one electron replaced by an antiproton.
- Antiprotonic lithium: an atom of lithium, with one electron replaced by an antiproton.
- Exciton: a bound state of an electron and an electron hole.
- Hypernuclear atoms: any of several observed atoms with a hypernucleus. Hypernuclei are any nuclei which contain (in addition to protons and neutrons) at least one hyperon, a subclass of baryons which contain strange quarks. These atoms are studied primarily for their nuclear behavior, and so fall better into the subfield of nuclear physics, rather than atomic physics or chemistry.
- Kaonic helium: a helium atom, with one electron replaced by a negative kaon, which is a meson composed of a strange quark, and an antiup quark.
- Kaonic hydrogen: a hydrogen atom, with the electron replaced by a negative kaon, a meson composed of a strange quark and an antiup quark.
- Kaonium: a bound state of a positive and negative kaon. Positive kaons are mesons composed of up and antistrange quarks, while negative kaons are mesons composed of a strange quark, and an antiup quark.
- Muonic helium: an atom of helium, with one electron replaced by a muon.
- Muonic hydrogen: an atom of hydrogen, with the electron replaced by a muon.
- Muonium: a bound state of a positive muon (also known an an antimuon) and an electron. There is also predicted to exist what is called “true muonium,” a bound state of a muon on an antimuon, but it has yet to be observed.
- Onium: this is the general term for the bound state of a particle with its own antiparticle. Pionium and positronium are examples.
- Pionic helium: an atom of helium, with one electron replaced by a negative pion. Pions are mesons, and the negative pion is composed of an up and an antidown quark.
- Pionic hydrogen: an atom of hydrogen, with one electron replaced by a negative pion, a meson composed of an up and an antidown quark.
- Pionium: a bound state of two pions, one positive and one negative. The negative pion is described above, and the positive pion, also a meson, is composed of a down and an antiup quark.
- Positronium: a bound state of a positron and an electron. This exotic atom can form an exotic molecule, together with a hydrogen atom; such an exotic molecule is called positronium hydride, and has the formula PsH. Another exotic molecule involving positronium is a bound state of two positronium atoms; it is called di-positronium. Positronium also forms halides and a cyanide.
- Protonium: a bound state of a proton and an antiproton.
- Quarkonium: a term for a meson which is the bound state of any quark and its own antiquark. While one can find examples in the literature where various forms of quarkonium are discussed as though they are exotic atoms, I prefer to view them simply as a subset of mesons, not a category of exotic atom.
- Sigmaonic atoms are thought to be possible, via such methods as replacing an electron in a hydrogen or helium atoms with a negatively-charged sigma baryon. However, I have found no evidence of actual observation of such particles.
- Tau-containing exotic atoms are predicted to occur, but have not been observed, yet, due to the short lifetime (less than a trillionth of a second) of the tau particle, a lepton. “Tauonium” is a term which has been used for these hypothetical exotic atoms.
M33, the Triangulum Galaxy, Adorning the Faces of a Pentagonal Icositetrahedron
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Evidence suggests that M33 is a satellite galaxy of the even better-known Andromeda Galaxy (M31), which happens to be on a collision course with our own Milky Way. In 1.5 billion years or so, Andromeda and the Milky Way will merge to form a giant elliptical galaxy already pre-named Milkomeda. At that point, the Triangulum Galaxy may become a satellite of Milkomeda (probably one of several), or be gravitationally ejected, or simply be absorbed into Milkomeda itself.
Here, it is projected on each face of the Catalan solid which is dual to the snub cube, using software you can try at http://www.software3d.com/Stella.php.
98-Faced Polyhedron Featuring Heptagons
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There are, as faces, 24 irregular heptagons, 6 irregular octagons of one type, and 12 of another, 24 rectangles of one type, and 24 of another, and 8 equilateral triangles. This was made using Stella 4d, which you may try or buy at http://www.software3d.com/Stella.php.
42-Faced Polyhedron Featuring Heptagons
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The blue faces are irregular heptagons, and are twenty-four in number. There are twelve of the green rhombi, and six of the red squares. This was made using Stella 4d, which you may try or buy at http://www.software3d.com/Stella.php.
I Started with Eight.
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Eight led to this.
If you think about eight long enough, you will understand.
News Flash: Big Snake on the Loose in Texas
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This just in from Longview, Texas — a five-meter (~16 foot) Burmese Python is on the loose.
If you see it, do not approach.
Retreat and dial 911.
No kidding.
Also: learn the metric system.
Circumslices
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Regions between close-packed circles of equal radius resemble triangles, but with 60 degree arcs replacing the sides. As these regions are the only things left of a plane after all such circles are sliced out, and they each are outside all the circles used, I’ve decided to name them “circumslices.” Interestingly, the three interior angles of a circumslice each asymptotically approach zero degrees, as one approaches circumslice-vertices, which are also the points of contact of the circles.
Why did I name these things “circumslices?” Because they needed a name, that’s why!
A Rotating Rhombic Dodecahedron with Rotating Tessellations on Its Faces
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Several recent posts here have been of tessellations I have made using Geometer’s Sketchpad and MS-Paint. To create this rotating polyhedron, I selected one of these tessellations, and projected it onto each face of a rhombic dodecahedron, using another program called Stella 4d. Unlike in the last, similar post, though, I set these tessellation-images to keep their orientation, from the point of view of a stationary observer watching the entire polyhedron rotate, from a distance. Since the polyhedron itself is rotating, this creates a rotation-effect for the tessellation-image on each face.
You can try Stella 4d for yourself, right here, for free: http://www.software3d.com.stella.php.
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