Greg WhiteheadFeb 23
ruud de rooij
The 19 ways to split a 2x2x2 cube into 2 equal tetracubes.
Feb 23 at 2:20amWeb
Show threadmeduz'Feb 23

@ruuddotorg Could it be that some of them are the same, but the overall results is rotated?

It looks like the top left one, the one at the rightest right and the one in the center of the bottom line are the same. 👀

Show threadThe Tiller 🌈Feb 23

@meduz @ruuddotorg
:+1:

If you fully account for mirroring and rotational symmetry, only 3 ways remain I think

Show threadRobynFeb 23
@swoonie
I would argue that rotational symmetry is the same division - the pieces are identical if rotated in relation to each other. Mirroring creates tetracubes that can't map onto each other. So 4 ways?
@meduz @ruuddotorg
Show threadruud de rooijFeb 24
@RedRobyn @swoonie @meduz That’s correct, considering rotations there are 4 distinct shapes, 2 of which are mirror images of each other.
Show threadPhosphenes4d ago

@ruuddotorg @RedRobyn @swoonie @meduz

I am trying to understand the rules...

The bottom middle one does not seem to have a rotated twin anywhere, but the bottom left one appears 3 times at different rotations. Why is this?

Show threadRobyn4d ago
@Phosphenes
The bottom middle one appears rotated immediately above and to the right of itself, to the far right of the middle row and also top left.
Show threadmeduz'4d ago
@Phosphenes @ruuddotorg @RedRobyn @swoonie It actually appears 3 times, too: https://m.nintendojo.fr/@meduz/116118750335048358
meduz' (@meduz@m.nintendojo.fr)

@ruuddotorg@hachyderm.io Could it be that some of them are the same, but the overall results is rotated? It looks like the top left one, the one at the rightest right and the one in the center of the bottom line are the same. 👀

NintendojoFR
Show threadRobyn4d ago
@meduz
Four times in all.
The fourth is the one diagonally above and to the right of the middle bottom.
Show threadPhosphenes3d ago

@RedRobyn @meduz

D'oh of course. I was seeing it wrong.

Show threadTikituFeb 23
@ruuddotorg I looked for symmetries in the layout and then realised there’s no privileged element to put in the center. Now I’m trying to figure out why 19, symmetry groups etc. Lovely stuff.
Show threadruud de rooijFeb 23
@tikitu There are 3 variants, one with 3 orientations (the “pancakes”), one with 4 orientations (the “y”) and one with 12 orientations (the “snakes”, actually 6 for each mirror image), for a total of 19.
Show threadMark GritterFeb 23
@ruuddotorg @tikitu It's neat that in this case we always get two equivalent pieces. For example in the decomposition of 4x4x4 cubes into 32-cube chunks I don't think this would be the case. We could start with two pancakes and then swap two cubes not related by a symmetry. (And of course for 3x3x3 same volume is impossible!)
Show threadMartyFeb 23
@ruuddotorg Nice work, this is great!
Show threadThe Tiller 🌈Feb 23
@ruuddotorg
In any case the graphic design is ace
Show threadrntzFeb 23

@ruuddotorg if you allow disconnected tetracubes you get 28 more possibilities, I think, made by three disconnected-tetracube-within-2x2-cube shapes:

(1) a 2 cube line and another parallel 2 cube line diagonally opposite it. 3 orientations (one per axis).

(2) three cubes in an L shape plus a disconnected cube diagonally opposite the corner of the L. 8 (choices of corner) * 3 (orientations of L) = 24 orientations, I think.

(3) four disconnected cubes. 1 orientation, I think.

Show threadrntzFeb 23

@ruuddotorg argh, I think you have to divide the 24 orientations for (2) by some symmetry factor. I'm guessing in half but need to double-check...

EDIT: yeah, I think it's 12. you can generate them uniquely by starting with 2 pancakes (3 options) and then choosing one of the four corners in in the plane of their orientation to swap between the two pancakes. 3 * 4 = 12.

if this is right, that's 12+3+1 = 16 disconnected tetracube decompositions.

Show threadRue MohrFeb 23
@ruuddotorg new tetris dropped?
Show threadJim SpathFeb 23

@ruuddotorg Reminds me of Piet Hein's Soma Cube!

https://piethein.com/shop/183-games-amp-puzzles/217-soma-cube-natural-oak-fsc-8x8x8-cm-invented-1933/

#3D

PIET HEIN SOMA-8*8*8 cm - wood

From time to time efforts have been made to devise a puzzle in three dimensions. None, in my opinion, has been as successful as the Soma cube, invented by...

Show threadAndy AlderwickFeb 23
@ruuddotorg beautiful 
Show threadxannaFeb 23
@ruuddotorg beautiful
Show threadStarxxonFeb 24

@ruuddotorg Oh I've played with these shapes before!

Here's some experiments I made trying to organize these shapes. Note that my project excluded the "snake" shapes.

(The second picture is a flat representation of the first one, with top and bottom layers of each shape as two 2x2 pixels bitmaps.)

Show threadFrankoFeb 24
@ruuddotorg That reminds me of my recruitment test for my apprenticeship as a model maker in 1993.
Show threadOrion Ussner kidder5d ago
@ruuddotorg Pretty.
Show threadJohn Carlsen 🇺🇸🇳🇱🇪🇺5d ago

@ruuddotorg

Could they be arranged in a sensible order in which the three with simple halves are 120 degrees apart?

Show threadSjoerd Visscher4d ago
@johnlogic @ruuddotorg I tried and it does seem to be possible to have nice symmetries in the layout!
Show threadSjoerd Visscher2d ago
@johnlogic @ruuddotorg Here it is as handwritten SVG, making use of symmetries so that it only has to dray one sixth of the whole image. https://codepen.io/sjoerdvisscher/full/azmdvqq
19 ways to split a 2x2x2 cube into 2 equal tetracubes

...

Show threadJohn Carlsen 🇺🇸🇳🇱🇪🇺2d ago

@sjoerd_visscher @ruuddotorg

Hey, that came out nice!

Show threadMartin Grider2d ago
@sjoerd_visscher @johnlogic @ruuddotorg This is gorgeous, and I think I'm going to print it out and frame it.
Show threadgroff 🇺🇦5d ago
@ruuddotorg I hate that it's 19.
Show threadMister Dave4d ago
@geoffl @ruuddotorg the diagram demonstrates why this is actually a beautiful number: it shows a symmetry between cubes in 3-space and a hexagonal tiling of the plane.
Show thread🐙ptoothfish🐙5d ago
@ruuddotorg the cuba sutra!
Show threadNegative12DollarBill5d ago
@ruuddotorg
Is there a formula for how many there are? Dimensions × faces plus one? Nineteen is unexpected.
Show threadruud de rooij5d ago
@negative12dollarbill Not sure if there’s a formula, but this post explains why it’s 19: https://hachyderm.io/@ruuddotorg/116120592821401426
Show threadErik Nelson5d ago
@ruuddotorg how many are there if you don't count rotations and reflections?
Show threadFlorine4d ago
@ruuddotorg nice, they remind me of the Fibonacci blocks I once made, especially the figuring out what shape I liked for every number
https://wisknutsels.wordpress.com/2012/12/10/alle-f-blokjes/
Alle F-blokjes

Hier zijn ze dan in hun volle glorie, alle F-blokjes die ik heb gemaakt, de nrs 0 t/m 9. De berichten over de blokjes vind je hier: 0 en 1, 2 en 3, 4 en 5, 6, nieuwe versies van 3,4 en 6, 7, 8, 9. …

Wisknutsels
Show threadLiu Yao 刘杳4d ago
@ruuddotorg Unrelated to this, how many ways to split a cube into tetrahedra (not necessarily of the same shape or volume, but the four vertices all come from the 8 vertices of the cube)? And in higher dimensions?