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Modeling the Reptigan Union's Space
Posted: Posted July 24th
Edited July 27th by chiarizio
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Each little bit of Reptigan Union Space could be a rhombal dodecahedron, an Archimedean solid with twelve congruent rhombuses for faces, each of them with angles of sixty and 120 degrees, like two congruent equilateral triangles stuck together at the bases.
The reason for that figure is, it’s nearly a sphere; and if I close-packed some space by a face-centered-cubic close-packing of equal spheres, then expanded all the spheres identically to take up the empty space between them, I’d have tiled the space with rhombal dodecahedrons.

Label the center of each such dodecahedron with an ordered quartet of three-digit non-negative whole numbers, from 000 to 999.
Limit them to three degrees of freedom by requiring their sum to be 1,998.
Then the six extreme points of Union Space will have these coordinates:
[999;999;000;000]
[999;000;999;000]
[999;000;000;999]
[000;999;999;000]
[000;999;000;999]
[000;000;999;999]

Adpihi, the human capital/Capitol of the Reptigan Union, is near the center;
let’s just say it’s at
[500;500;499;499]

A given rhombal-dodecahedron “tile” will have twelve nearest-neighbors.
For instance [w;x;y;z] would have the following nearest-neighbors.
[w+1;x-1;y;z]
[w+1;x;y-1;z]
[w+1;x;y;z-1]
[w-1;x+1;y;z]
[w;x+1;y-1;z]
[w;x+1;y;z-1]
[w-1;x;y+1;z]
[w;x-1;y+1;z]
[w;x;y+1;z-1]
[w-1;x;y;z+1]
[w;x-1;y;z+1]
[w;x;y-1;z+1]

In other words, add one to one coordinate, and subtract one from another coordinate.

If one of a tile’s coordinates is 000 you can’t subtract one, and if one of them is 999 you can’t add one. So tiles on the boundary will have fewer than twelve nearest neighbors in Union Space.

My battery is running low. I sure as hell hope I can continue this tomorrow morning!



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There are 666,667,000 rhombal-dodecahedron “tiles” in Union Space.
Each has a diameter of about a light-year.
The distance between two of them can be calculated from their coordinates this way;
Suppose we want the distance between [a;b;c;d] and [w;x;y;z]
Take all the differences between corresponding coordinates;
a-w, b-x, c-y, and d-z.
Square each of those differences;
(a-w)^2, (b-x)^2, (c-y)^2, and (d-z)^2.
Sum all those squared differences;
Divide that sum by two (2);
And take the square root of that quotient.
The result will be the distance in light-years.
Each tile’s center will be about a light-year from the center of each of its nearest neighbors.
Each tile will be about a light-year across. (Ie in diameter).


Edited July 28th by chiarizio
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Union space itself will be about 999*squrt(2) light-years from each extreme point to the furthest opposite extreme point.

Almost all of the tiles will contain nothing.



Posted July 24th by chiarizio
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I was told long ago, and accepted as a working hypothesis for years, that in parts of our galaxy like Sol’s “neighborhood”, the average distance between a star-system and the nearest neighboring star-system was about 10 or 11 light-years.
But now Wikipedia says it’s actually about 5 light-years.

So I actually went and counted* the star-Systems within 500 light-years from Sol; and it works out to a density of about one star-system for every 3029 or 3030 cubic light-years. That’s consistent with an average distance between each star-system and its nearest neighboring star-system of a shade more than 8.9 light-years.
*really? That’s like 172,800 star-systems ! I must have taken a shortcut of some kind!

Anyway I figure Union space will be similar; about one star-system for every 3000-or-so cubic light-years, and about 9-or-so light-years between a star-system and its nearest-neighbor star-system.

For computer gaming-or-simulation-or-database-maintenance about Reptigan space, it makes better sense to keep a list of star-systems and record the coordinates of each of them, than to keep a three-dimensional matrix of 666,667,000 “tiles” and list the contents of each tile in the matrix. The matrix is very sparse because so much of space has no star-stuff in it.

—————


—————

Stars’ tendencies to be in multi-star systems depend on their spectral class, surface temperature, and mass, all of which are correlated.

—————

The mean and median and mode number of planets a star has is probably in the range of two to six.
The mean and median and mode number of planets-in-the-Goldilocks-Zone is probably in the range one to three.
These last two statements might apply especially to class M stars.

—————

The heaviest F stars — the F0s —— and the lightest A stars — the A9s —— are right at Chandrasekhar’s limit. (1.38 to 1.44 Solar masses).
They might, and anything heavier will, go supernova eventually.

—————

———





Edited August 11th by chiarizio
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I was told long ago, and accepted as a working hypothesis for years, that in parts of our galaxy like Sol’s “neighborhood”, the average distance between a star-system and the nearest neighboring star-system was about 10 or 11 light-years. But now Wikipedia says it’s actually about 5 light-years.


Out of curiosity was the 10ly hypothesis counting only easily visible (I guess main line?) stars, or dwarf were dwarf stars included as well? There aren't many OBAFGK stars within about 16ly from the Sun (I count 10); and, apart from Centauri, the next closest are 9 and 10ly away. Whereas dwarf stars are in the 4, 6, 7 and 8ly ranges.
https://www.space.com/18964-the-nearest-stars-to-earth-infographic.html
If Nemesis turns out to be an actual star, it would likely be a dwarf of some colour and also rather closer than either 5 or 10ly distance. ;)


Posted July 24th by elemtilas
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@elemtilas:

Out of curiosity was the 10ly hypothesis counting only easily visible (I guess main line?) stars, or dwarf were dwarf stars included as well? There aren't many OBAFGK stars within about 16ly from the Sun (I count 10); and, apart from Centauri, the next closest are 9 and 10ly away. Whereas dwarf stars are in the 4, 6, 7 and 8ly ranges.


I suspect there’s some such explanation.
I’m sure they included yellow dwarfs, and probably also most of the orange dwarfs we know about now. And they included the red dwarfs they knew about; but they probably knew about many fewer red dwarfs then than they know about know.
They didn’t know about brown dwarfs, so they didn’t include any of them. I’m not intending to include them either; at least not in my posts up til now. Our model may have lots of brown dwarfs, but I doubt brown-dwarf systems are going to have permanent populations in the myriads.

https://www.space.com/18964-the-nearest-stars-to-earth-infographic.html


Thanks!





Posted July 25th by chiarizio
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I doubt brown-dwarf systems are going to have permanent populations in the myriads.


They might be a garden spot, for some kinds of folks!

Posted July 25th by elemtilas
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@Xhin:
I can access this thread if not logged in. I can’t access it if logged in.

They might be a garden spot, for some kinds of folks!


@elemtilas:
What do you mean?

Posted July 25th by chiarizio
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I can access this thread now when logged on.
I still can’t fix the apostrophe thing.


Posted July 25th by chiarizio
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I don’t expect a binary star with a planet or planets orbiting just one of its stars in the Goldilocks zone, to also have a planet orbiting the entire pair in the pair’s Goldilocks zone.
Either any orbit close enough to either star to orbit just that star rather than the pair, is too hot for water to remain liquid, instead of boiling away;
or any orbit distant enough from both stars to orbit them both as a pair, is too cold for water to remain liquid, instead of freezing;
or there is no stable Goldilocks orbit.

But I do want people (human and/or non-human) to settle the systems of solitary F and G and K and M stars,
and to settle in the systems of binary FF FG FK FM GG GK GM KK KM MM star-pairs.

I’m going to assume that life in a three-or-more star-system is too complex and dangerous for a large permanent settlement of intelligent inhabitants.
But maybe there’ll be small-to-medium quasi-permanent research stations etc. there, near some of them, that occasionally have to be temporarily shut down and evacuated because of stellar weather, and that nobody raises a family in.

The same might be true of some systems containing one or two class A stars.



Edited July 25th by chiarizio
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They might be a garden spot, for some kinds of folks!

What do you mean?




Worlds orbiting brown dwarf stars might be rather appealing to some folks...



Posted July 25th by elemtilas
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