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Tom Tyson
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Tom Tyson

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"And anyway, young Kirby, I was flying rockets when you were kicking in your cradle."
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Volcanic explosion on Io: Image by Voyager 1. Distance to Io was about 490,000 kilometers (304,000 miles). An enormous volcanic explosion can be seen silhouetted against dark space over Io's bright limb. At this time solid material had been thrown up to an altitude of about 100 miles.

/NASA/JPL/
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Tom Tyson

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As you likely know, the Hugo Awards were announced yesterday. I was invited to be on the Hugo-nominated Coode Street Podcast for their annual Hugo ballot rundown with hosts Jonathan Strahan and Gary K. Wolfe, as well as guest Tansy Rayner Roberts. That…
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What lights up the Flame Nebula?
Fifteen hundred light years away in Orion lies a nebula which, from its glow and dark dust lanes, appears like a billowing fire. But fire, the rapid acquisition of oxygen, is not what makes this Flameglow. Rather the bright Alnitak, in the Hunter's belt visible to the nebula's right, shines energetic light into the Flame that knocks electrons away from the great clouds of hydrogen gas that reside there. Much of the glow results when the electrons and ionized hydrogen recombine. The Flame Nebula (NGC 2024) is part of the Orion Molecular Cloud Complex, a star-forming region that includes the Horsehead Nebula.

R.A : 05h 41m 54s
Dec. : -1° 51′ 0.0"
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Want to be a superhero of clever timing? Why not treat your players to an original X-Men RPG adventure - Nightmares of Future Past.
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I remember that series...ran a mini campaign using some of the material back in the day.
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Tom Tyson

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"All characters start at 0 level.  Most will die in a dungeon, alone and unknown."
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Yes, was browsing it earlier.
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The small stellated dodecahedron

What if you let yourself make shapes that are as symmetrical as Platonic solids, but where all the faces are stars?   Then you get things like this.

If you look carefully, you'll see lots of 5-pointed stars.  Each one is a regular pentagram - a 5-pointed star whose corners are a regular pentagon.  Each one touches 5 others at each corner, in exactly the same way.  So, it's as regular as you might want. 

But it's funny in some ways.  First, the faces are stars instead of regular polyhedra.  Second, the faces intersect each other: that's why you don't see all of any star. 

There are 4 polyhedra whose faces are all regular stars, with each face just like every other and each vertex like every other.  They're called Kepler-Poinsot polyhedra

This particular one is called the small stellated dodecahedron because if you remove all the pyramid-shaped pieces you're left with a dodecahedron!  Each star lies in the same plane as one of this dodecahedron's faces.  So, there are 12 stars in this shape.

On the other hand, the sharp points of this shape form the corners of an invisible icosahedron!  So, there are 20 sharp points.

Puzzle: how many edges does this shape have?

This shape can be seen in a floor mosaic in the Basilica of St. Mark in Venice, built in 1430.  It was rediscovered by Kepler in his work Harmonice Mundi in 1619.  This book, about the "harmonies of the world", is an amazing mix of geometry, astronomy and music theory - a mystical warmup for his later breakthroughs on the orbits of the planets. 

Much later, Escher made himself a wood model of the small stellated dodecahedron, which he drew in two woodcuts called Order and Chaos.

While the Kepler-Poinsot polyhedra are beautiful, I've avoided studying them because I don't see how they fit into the theory of Coxeter groups - the study of discrete symmetries that connects Platonic solids, Archimedean solids and hyperbolic honeycombs to deeper strands of math like Lie theory, the study of continuous symmetries.  I've been afraid these shapes are merely cute, not deep.

Maybe it's time to find out.

For more, see:

http://mathworld.wolfram.com/SmallStellatedDodecahedron.html

https://en.wikipedia.org/wiki/Small_stellated_dodecahedron

The Mathworld page has a much better picture of the mosaic in the Basilica of St. Mark.

#geometry  
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This is inside my d20s! How awesome is that!
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Humanity never fails to impress me.
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    Sandship Captain, 1994 - 1996
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