A Polyhedral Journey, Beginning with Face-Based Zonohedrification of an Icosahedron

To begin this, I took an icosahedron, and made a zonish polyhedron with it, with the new faces based on the zones of the existing faces. Here’s the result.

1 face-based zonish icosahedron

Next, I started stellating the polyhedron above. At the sixth stellation, I found this. It’s a true zonohedron, and the first polyhedron shown here is merely “zonish,” because one has triangles, and the other does not. (One of the requirements for a polyhedron to be a zonohedron is that all its faces must have an even number of sides.)

2 6th stellation face-based zonish icosahedronAfter that, I kept stellating, finding this as the 18th stellation of the first polyhedron shown here.

3 18th stellation face-based zonish icosahedron

With this polyhedron, I then made its convex hull.

4 Convex hull of 18th stellation of face-based zonish icosahedronAt this point, the irregular hexagons were bothering me, so I used Stella 4d‘s “try to make faces regular” option. (Stella 4d is polyhedron-manipulation software you can try for free, or purchase, right here.)

5 spring model of convex hull of 18th stellation of face-based icosahedron

The next step I chose was to augment all the yellow trapezoids with prisms, each with a height 1.6 times the trapezoids average edge length.

6 Augmented sping model of convex hull of 18th stellation of FBZI

The next step was, again, to make the convex hull.

7 Convex hull of augmented convex hull

At this point, I tried “try to make faces regular” again, and was pleased with the result. The green rectangles became so thin, however, that I had to stop displaying the edges and vertices, in order for then to be seen.

8 spring model of last oneNext, I augmented both the blue faces (decagons) and the yellow faces (dodecagons) with antiprisms, again using a height 1.6 times that of the augmented faces’ average edge-lengths.

9 Augmented Poly 9th in series

Next, I made the convex hull again — a step I often take immediately after augmenting a polyhedron.

10 Convex hull

This one surprised me, as it is more complicated than I expected. To clean things up a bit, I augmented only the trapezoids (dark pink) with prisms, and dodecagons (green) with antiprisms, again using the factor 1.6 for the augmentation-height.

11 augmented Convex hull

The next step I chose was to take the convex hull, once more. I had not yet noticed that the greater height of the trapezoidal prisms would cause the dodecagonal antiprisms to be “lost” by this step, though.

12 convex hull

Next, “try to make faces regular” was used again.

13 spring model

This last result had me feeling my polyhedral journey was going in circles, so I tried augmentation again, but in a different way. I augmented this polyhedron, using prisms, on only the red trapezoids (height factor, 1.6 again) and the blue rectangles (new height factor, 2.3 times average edge length).

14 augmented spring model

After that, it was time to make another convex hull — and that showed me that I had, indeed, taken a new path.

15 Convex hullI found the most interesting faces of this polyhedron to be the long, isosceles trapezoids, so I augmented them with prisms, ignoring the other faces, using the new height-factor of 2.3 times average edge length this time.

16 augmented Convex hull

Of course, I wanted to see the convex hull of this. Who wouldn’t?

17 Convex hull

I then started to stellate this figure, choosing the 14th stellation as a good place to stop, and making the edges and vertices visible once more.

18 the 14th stellation of the previous Convex hull

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A Zonish Icosahedron, and Some of Its “Relatives”

To begin this, I used Stella 4d (available here) to create a zonish polyhedron from the icosahedron, by adding zones along the x-, y-, and z-axes. The result has less symmetry than the original, but it is symmetry of a type I find particularly interesting.

zonohedrified icosahedron xyz

After making that figure, I began stellating it, and found a number of interesting polyhedra in this polyhedron’s stellation-series. This is the second such stellation:

zonohedrified icosahedron xyz 2nd stellation

This is the 18th stellation:

zonohedrified icosahedron xyz 18th stellation

The next one, the 20th stellation, is simply a distorted version of the Platonic dodecahedron.

zonohedrified icosahedron xyz 20th stellation

This one is the 22nd stellation:

zonohedrified icosahedron xyz 22nd stellation

This is the 30th stellation:

zonohedrified icosahedron xyz 30th stellation

The next really interesting stellation I found was the 69th:

zonohedrified icosahedron xyz 69th stellation

At this point, I returned to the original polyhedron at the top of this post, and examined its dual. It has 24 faces, all of which are quadrilaterals.

zonohedrified icosahedron xyz dual

This is the third stellation of this dual — and another distorted Platonic dodecahedron.

zonohedrified icosahedron xyz dual 3rd stellation

This is the dual’s 7th stellation:

zonohedrified icosahedron xyz dual 7th stellation

And this one is the dual’s 18th stellation:

zonohedrified icosahedron xyz dual 18th stellation

At this point, I took the convex hull of this 18th stellation of the original polyhedron’s dual, and here’s what appeared:

Convex hull of 18th stellation of dual of zonish icosahedron xyz

Here is this convex hull’s dual:

dual of Convex hull of 18th stellation of dual of zonish icosahedron xyz

Stella 4d, the program I use to make these (available here), has a built-in “try to make faces regular” function. When possible, it works quite well, but making the faces of a polyhedron regular, or even close to regular, is not always possible. I tried it on the polyhedron immediately above, and obtained this interesting result:

spring model of Dual of convex hull of stellation of zonish xyz icosahedron

While interesting, this also struck me as a dead end, so I returned to the red-and-yellow convex hull which is the third image above, from right here, and started stellating it. At the 19th stellation of this convex hull, I found this:

19th stellation of Convex hull of 18th stellation of dual of zonish icosahedron xyz

I also found an interesting polyhedron as the 19th stellation of the dual which is three images above:

19th stellation of dual of Convex hull of 18th stellation of dual of zonish icosahedron xyz

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Cuboctahedral Cluster of Rhombic Triacontahedra

Augmented Rhombic Triaconta

Due to their high number of planes of symmetry, rhombic triacontahedra make excellent building blocks to build other polyhedra. To make this, I used a program called Stella 4d, which you can try right here.

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Sam Harris: A Quote

Sam_Harris_01

Source — Waking Up:  A Guide to Spirituality Without Religion, p. 38.

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Anticarbon-14 and Oxygen-18 Nuclei: What If They Collided? And Then, What About the Reverse-Reaction?

anticarbon-14 and oxygen-18Were nuclei of anticarbon-14 and oxygen-18 to collide (and their opposite charges’ attractions would help with this), what would happen? Well, if you break it down into particles, the anticarbon-14 nucleus is composed of six antiprotons and eight antineutrons, while the oxygen-18 contains eight protons and ten neutrons. That lets six proton-antiproton pairs annihilate each other, releasing a specific amount of energy, in the form of gamma rays, with that amount calculable using E=mc² and KE=½mv². The two excess protons from oxygen-18, however, should escape unscathed. In the meantime, eight neutron-antineutron pairs also are converted into a specific, calculable amount of gamma-ray energy, but with two neutrons surviving. Here’s the net reaction:

particles

Two protons and two neutrons, of course, can exist as separate particles, two deuterons, a tritium nucleus and a neutron, or a single alpha particle.

Now, consider this:  any physical process is, at least hypothetically, reversible. Therefore, it should be possible to bombard a dense beam of alpha particles with many gamma rays, each of a specific and calculable energy, and, rarely, the reverse reaction would occur, and anticarbon-14 and oxygen-18 nuclei would appear. Oxygen-18 is stable, but rare, so detection of it would be evidence that the reverse-reaction had occurred. Anticarbon-14, however, can logically being expected to decay to antinitrogen-14 via the antimatter version of beta-negative decay, which, it being antimatter, will result in the emission of an easily-detectable positron. It likely will not have time to do this, though, for carbon-14’s half-life (and anticarbon-14’s as well, one assumes) exceeds 5,000 years. The more likely scenario for the anticarbon-14 nucleus is that it will create a large burst of gamma rays when it encounters, say, a non-antimatter carbon atom — and these gamma rays would come from a different position than the ones bombarding the alpha particles, and can therefore be distinguished from them by determination of their direction.

Such a reverse-reaction would be quite rare, for it involves a decrease in entropy, violating the Second Law of Thermodynamics. However, the Second Law is a statistical law, not an absolute one, so it simply describes what happens most of the time, allowing for rare and unusual aberrations, especially on the scale of things which are extremely small. So, do this about a trillion times (or much more, but still a finite number of trials) and you’ll eventually observe evidence of the production of the first known anticarbon nucleus.

Also, before anyone points this out, I am well aware that this is highly speculative. I do make this claim, though:  it can be tested. Perhaps someone will read this, and decide to do exactly that. I’d test it myself, but I lack the equipment to do so.

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Top 100 Banned/Challenged Books: 2000-2009

censorship

The best ways to celebrate Banned Books Week (which is going on now) are to read/buy/give away banned books, and/or donate money to libraries which deliberately put banned books in the circulating collection, as all good libraries do.

I’ve color-coded the list below. Books in red, I have read in their entirety. Books in blue, I have read some of, but have not (yet) finished. Also, now that I know they’re on this list, I’m likely to add some of the books in black, which I have not yet read, to my “books-to-read” list. There are few things I hate as much as censorship.

1. Harry Potter (series), by J.K. Rowling

2. Alice series, by Phyllis Reynolds Naylor

3. The Chocolate War, by Robert Cormier

4. And Tango Makes Three, by Justin Richardson/Peter Parnell

5. Of Mice and Men, by John Steinbeck

6. I Know Why the Caged Bird Sings, by Maya Angelou

7. Scary Stories (series), by Alvin Schwartz

8. His Dark Materials (series), by Philip Pullman

9. ttyl; ttfn; l8r g8r (series), by Lauren Myracle

10. The Perks of Being a Wallflower, by Stephen Chbosky

11. Fallen Angels, by Walter Dean Myers

12. It’s Perfectly Normal, by Robie Harris

13. Captain Underpants (series), by Dav Pilkey

14. The Adventures of Huckleberry Finn, by Mark Twain

15. The Bluest Eye, by Toni Morrison

16. Forever, by Judy Blume

17. The Color Purple, by Alice Walker

18. Go Ask Alice, by Anonymous

19. Catcher in the Rye, by J.D. Salinger

20. King and King, by Linda de Haan

21. To Kill A Mockingbird, by Harper Lee

22. Gossip Girl (series), by Cecily von Ziegesar

23. The Giver, by Lois Lowry

24. In the Night Kitchen, by Maurice Sendak

25. Killing Mr. Griffen, by Lois Duncan

26. Beloved, by Toni Morrison

27. My Brother Sam Is Dead, by James Lincoln Collier

28. Bridge To Terabithia, by Katherine Paterson

29. The Face on the Milk Carton, by Caroline B. Cooney

30. We All Fall Down, by Robert Cormier

31. What My Mother Doesn’t Know, by Sonya Sones

32. Bless Me, Ultima, by Rudolfo Anaya

33. Snow Falling on Cedars, by David Guterson

34. The Earth, My Butt, and Other Big, Round Things, by Carolyn Mackler

35. Angus, Thongs, and Full Frontal Snogging, by Louise Rennison

36. Brave New World, by Aldous Huxley

37. It’s So Amazing, by Robie Harris

38. Arming America, by Michael Bellasiles

39. Kaffir Boy, by Mark Mathabane

40. Life is Funny, by E.R. Frank

41. Whale Talk, by Chris Crutcher

42. The Fighting Ground, by Avi

43. Blubber, by Judy Blume

44. Athletic Shorts, by Chris Crutcher

45. Crazy Lady, by Jane Leslie Conly

46. Slaughterhouse-Five, by Kurt Vonnegut

47. The Adventures of Super Diaper Baby: The First Graphic Novel by George Beard and Harold Hutchins, the creators of Captain Underpants, by Dav Pilkey

48. Rainbow Boys, by Alex Sanchez

49. One Flew Over the Cuckoo’s Nest, by Ken Kesey

50. The Kite Runner, by Khaled Hosseini

51. Daughters of Eve, by Lois Duncan

52. The Great Gilly Hopkins, by Katherine Paterson

53. You Hear Me?, by Betsy Franco

54. The Facts Speak for Themselves, by Brock Cole

55. Summer of My German Soldier, by Bette Green

56. When Dad Killed Mom, by Julius Lester

57. Blood and Chocolate, by Annette Curtis Klause

58. Fat Kid Rules the World, by K.L. Going

59. Olive’s Ocean, by Kevin Henkes

60. Speak, by Laurie Halse Anderson

61. Draw Me A Star, by Eric Carle

62. The Stupids (series), by Harry Allard

63. The Terrorist, by Caroline B. Cooney

64. Mick Harte Was Here, by Barbara Park

65. The Things They Carried, by Tim O’Brien

66. Roll of Thunder, Hear My Cry, by Mildred Taylor

67. A Time to Kill, by John Grisham

68. Always Running, by Luis Rodriguez

69. Fahrenheit 451, by Ray Bradbury

70. Harris and Me, by Gary Paulsen

71. Junie B. Jones (series), by Barbara Park

72. Song of Solomon, by Toni Morrison

73. What’s Happening to My Body Book, by Lynda Madaras

74. The Lovely Bones, by Alice Sebold

75. Anastasia (series), by Lois Lowry

76. A Prayer for Owen Meany, by John Irving

77. Crazy: A Novel, by Benjamin Lebert

78. The Joy of Gay Sex, by Dr. Charles Silverstein

79. The Upstairs Room, by Johanna Reiss

80. A Day No Pigs Would Die, by Robert Newton Peck

81. Black Boy, by Richard Wright

82. Deal With It!, by Esther Drill

83. Detour for Emmy, by Marilyn Reynolds

84. So Far From the Bamboo Grove, by Yoko Watkins

85. Staying Fat for Sarah Byrnes, by Chris Crutcher

86. Cut, by Patricia McCormick

87. Tiger Eyes, by Judy Blume

88. The Handmaid’s Tale, by Margaret Atwood

89. Friday Night Lights, by H.G. Bissenger

90. A Wrinkle in Time, by Madeline L’Engle

91. Julie of the Wolves, by Jean Craighead George

92. The Boy Who Lost His Face, by Louis Sachar

93. Bumps in the Night, by Harry Allard

94. Goosebumps (series), by R.L. Stine

95. Shade’s Children, by Garth Nix

96. Grendel, by John Gardner

97. The House of the Spirits, by Isabel Allende

98. I Saw Esau, by Iona Opte

99. Are You There, God?  It’s Me, Margaret, by Judy Blume

100. America: A Novel, by E.R. Frank

Source:  http://www.ala.org/bbooks/top-100-bannedchallenged-books-2000-2009

Finally, what I am reading, myself, during Banned Books Week is Sam Harris’s latest, Waking Up. It’s a safe bet that all books by Sam Harris are banned in quite a few places.

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The 109th Stellation of the Triakis Icosahedron

109th stellation of Triakis icosa

Created using Stella 4d:  Polyhedron Navigator, available here.

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