Sol LeWitt, 1974 · counted again, in your browser
The Same Cube, Turned
In 1974 Sol LeWitt counted the ways to take edges off a cube and leave a frame that, by the rules his list obeys, is one piece, stands in all three dimensions, and is not just another frame turned around. He found 122. Flick a rule and your browser recounts all 4,096 ways to choose which of the 12 edges to keep. The count is right, but his chart is not quite: as Natasha Rozhkovskaya and Michael Reb reported, its 10/4 and 10/5 are one cube turned, and that cube's mirror twin is missing.
The rules his list obeys
A cube has 12 edges. Keep some and take the rest away, and there are 4,096 ways to choose, from keeping none to keeping all of them. Sol LeWitt's Variations of Incomplete Open Cubes, shown at the John Weber Gallery in New York in 1974, is his list of the ones that count: an installation of 122 frames, each a cube with edges taken away, and a chart of drawings that labels them by how many edges they keep, from 3/1 to 11/1.
This page found no statement of the rules in LeWitt's own words. The mathematicians who have studied his list state them the same way: Natasha Rozhkovskaya and Michael Reb in a preprint of 2014 (published in 2015), Reb in his Master of Science report of 2013, and Mikael Vejdemo-Johansson in 2026. In Rozhkovskaya and Reb's words:
- One piece
the structure should be connected.
- Three-dimensional
a square, or just one edge are not listed in the project, since they are not three-dimensional.
- Turned is the same, mirrored is not
two structures are considered to be identical, if one can be transformed into another by a space rotation (but not by a reflection).
- Not the whole cube An incomplete cube lacks at least one edge. Vejdemo-Johansson asks for a
connected and non-planar proper and non-empty subset
of the edges.
Four switches, recounted as you flick them
122 frames under these rules. LeWitt's chart has 122, and every row matches.
Frames by number of edges kept (a dot is none)
| edges | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| these rules | · | · | · | 3 | 5 | 14 | 24 | 32 | 25 | 13 | 5 | 1 | · |
| his chart | · | · | · | 3 | 5 | 14 | 24 | 32 | 25 | 13 | 5 | 1 | · |
Every frame the switches allow is drawn here by this page, in its own order: by number of edges, mirror twins side by side, then the frames that are their own mirror image. Pick one to see it larger and turn it.
How it counts: each of the 4,096 subsets is tested against the switches, and each one that passes is turned all 24 ways (all 48, when a mirror image is the same frame) to find its smallest form; subsets with the same smallest form are one frame.
His chart's rows run 3/1 to 3/3, 4/1 to 4/5, 5/1 to 5/14, 6/1 to 6/24, 7/1 to 7/32, 8/1 to 8/25, 9/1 to 9/13, 10/1 to 10/5 and 11/1: that is 3, 5, 14, 24, 32, 25, 13, 5 and 1 drawings, 122 in all. Of the 24 ways to set the four switches, only one reproduces those rows, and it is the setting you started with. Count a mirror image as the same frame and the rows would be 2, 3, 9, 14, 19, 16, 9, 4 and 1, 77 in all. So whatever LeWitt did or did not write down, his count treats a frame and its mirror image as two frames: the tallies show it.
The same nine numbers are sequence A393969 in the On-Line Encyclopedia of Integer Sequences, entered by Paolo Xausa in 2026.
Two frames no turn can swap
The smallest row already shows why mirror images matter. With 3 edges there are 3 frames: three edges meeting at one corner, and two staircases that each take one step in every direction. The two staircases are mirror images, and no way of turning the cube carries one onto the other, so the list needs both. Across the 122 frames the rules allow, 45 pairs are mirror twins and 32 frames are their own mirror image: 45 + 45 + 32 = 122, and merging the twins leaves 45 + 32 = 77.
The economical count
A computer does all of this in a blink, which recalls an old contrast in writing about the work. Rosalind Krauss, in her essay LeWitt in Progress (published in the journal October in 1978), set his serial lists against algebra: The babble of a LeWitt serial expansion has nothing of the economy of the mathematician’s language.
The words are quoted from the excerpt of the essay on SOCKS.
Here is that economy. Burnside's lemma counts the frames without listing a single one. Take each of the 24 ways to turn a cube, count the subsets of edges that follow the rules and that this turn leaves exactly as they were, and average.
Burnside's lemma, for the switches above
| kind of turn | how many | rule-following subsets each leaves unmoved | together |
|---|---|---|---|
| leave it as it is | 1 | 2,366 | 2,366 |
| quarter turn about the line through two face centres | 6 | 2 | 12 |
| half turn about the line through two face centres | 3 | 22 | 66 |
| third of a turn about a long diagonal | 8 | 11 | 88 |
| half turn about the line through two edge midpoints | 6 | 66 | 396 |
| all turns | 24 | 2,928 |
2,928 ÷ 24 = 122
One line of arithmetic, and it is right. It is also blind. It says how many frames there are; it cannot say whether a particular set of 122 drawings is the right set. Draw one frame twice and leave out its mirror twin, and the chart still has 122 cells, and every row still has its tally.
The 10-edge row
Keep 10 edges and you have taken 2 away. The two missing edges can meet at a corner, lie parallel on one face, lie parallel at diagonally opposite edges of the cube, or be skew: neither meeting nor parallel. A skew pair comes in two hands, mirror images of each other, so the row has 5 frames, not 4.
His row against the computed row
- two edges at one corner
- two parallel edges on one face
- two diagonally opposite edges
- two skew edges, one hand
- two skew edges, the other hand not on his chart
- 10/1 parallel, on one face
- 10/2 diagonally opposite
- 10/3 at one corner
- 10/4 skew, one hand same
- 10/5 skew, the same hand same
The faint dotted lines are the two missing edges. 10/4 and 10/5 look different from here. Press Turn.
This is Rozhkovskaya and Reb's finding. Classifying the whole list, they confirmed the count, the artist found the correct number of structures (that is 122), however his list contains a mistake in the presentation of a pair of incomplete cubes
, and they located it: the artist mistakenly put the same incomplete open cube twice (the second time rotated), instead of two structures that are mirror images of each other
, at 10/4 and 10/5. Two turns of the cube carry 10/5 onto 10/4: a third of a turn about a long diagonal, or a half turn about the line through two edge midpoints. The other hand of the skew pair, the frame that belongs in one of those two cells, is not drawn.
So the chart has 122 cells but shows at most 121 different cubes, and the number 122 is still right. The count and the list are two different claims, and only the second needed checking cube against cube.
What this page checked, and what it did not
- It read the five drawings of the 10-edge row from a reproduction of the 1974 poster, recording only which two edges each cell lacks, and found what the computed row above shows: four different frames in five cells, with 10/4 and 10/5 the same. The diagrams of 10/4 and 10/5 that Rozhkovskaya and Reb print in their Figure 15, as LeWitt presented them, read the same.
- It has not checked the other 117 drawings one by one. Rozhkovskaya and Reb report a mistake in one pair; beyond its row tallies, this page makes no claim about the rest of the chart.
- A drawing like his puts the nearest and the farthest corner of the cube on the same point, so on its own it cannot say which hand a frame has: read the other way, every drawing becomes its mirror image. Read either way, as long as the whole row is read the same way, 10/4 and 10/5 are one frame and the other hand is missing.
- Rozhkovskaya and Reb write that LeWitt made small three-dimensional models to check that no frame was repeated, and, citing the 2001 Wadsworth Atheneum catalogue (which this page did not read), that he later asked two mathematicians to check the list,
who confirmed to the artist that the number of structures is 122
. Whether the frames themselves repeat the slip, this page does not know.
Numbers you may meet elsewhere
- 4,094. Rob Weychert's Incomplete Open Cubes Revisited (2018)
has found and rendered a total of 4,094 variations (including LeWitt’s original 122)
. That is every subset of the 12 edges except none and all, 4,096 − 2, with no turning: a different question, not a correction. - 135. Weychert describes the cubes this way:
each part is connected to at least one other part
. That is true of every frame on the list, but as a rule it is weaker than one piece. Set the first switch to each edge touches another and 13 more frames appear, 135 in all, every one of them in two separate pieces. Alison Pitt's 2023 re-enumeration states the test asdoes each edge touch at least one other edge?
, then traces whole paths through each frame, and she writes:I was left with exactly 122 configurations, the same as LeWitt’s.
- 2,366. The subsets of the edges that follow the rules before any turning is allowed: the first line of the Burnside table.
- Not flat. Vejdemo-Johansson reads three-dimensional as non-planar (the corners do not all lie in one plane); the switch above uses all three directions instead. For frames in one piece the two readings never disagree, so both give 122. They part once the one-piece rule is loosened: 150 instead of 135 for each edge touches another, and 210 instead of 175 with no rule.
Beyond the cube
Vejdemo-Johansson applied the same rules to all five Platonic solids and reports 6 tetrahedra, 122 cubes (just like LeWitt), 185 octahedra, 2 423 206 dodecahedra and 16 096 166 icosahedra
. Your browser recounts the two small ones the same way it counts the cube: 6 tetrahedra (6 edges, 12 turns) and 185 octahedra (12 edges, 24 turns), as he reports. The dodecahedron and the icosahedron have 30 edges each, so 1,073,741,824 subsets each; this page quotes those two counts and does not check them.
The check
The verifier rebuilds the cube from scratch in plain Node: its 12 edges, its 24 turns and 48 symmetries generated from two quarter turns and one mirror, connectivity by union-find, and every count by Burnside's lemma, a different route from the smallest-form sorting this page uses. It asserts the 4,096 subsets and the 2,366 that follow the rules; the total and every row tally for all 24 settings of the switches (122, 77, 135, 175, 126, 123 and 218 among them) and that only one setting matches the chart; every line of the Burnside table; the 45 twin pairs and 32 self-mirror frames; the five kinds of 10-edge frame; that 10/4 and 10/5 as read are one frame under either reading of the drawings, and that the corrected 10/5 of Rozhkovskaya and Reb is its mirror image; the counts for the tetrahedron and the octahedron; every figure on this page, sentence by sentence, with a sweep that fails on any figure left unasserted; and every quotation against its source passage in quotes.json, published beside this page (inside the site's own repository, also against a kept copy of each source).
Free choices and uncertainties:
- The rules are as these writers state them; none of them quotes LeWitt stating them, and this page found no statement of them in his own words. The rotation-only convention is shown by his row tallies, not by a quotation from him.
- Three-dimensional means using all three directions; the not-flat reading gives the same 122. Each edge touches another turns away a lone edge, and the empty frame with no edges at all is counted only when the one-piece and three-dimensional switches are both set to no rule.
- The chart's row tallies, and the two missing edges of each 10-edge cell, were read by eye from one reproduction of the poster, Figure 1 of Vejdemo-Johansson's paper. The verifier checks what follows from that reading, not the reading itself.
- No other row of the chart has been checked drawing by drawing here.
Run it yourself (Node 18 or later, nothing to install):
curl -sO https://artwaste.land/checks/verify-the-same-cube-turned.mjs
node verify-the-same-cube-turned.mjsSources
- Natasha Rozhkovskaya and Michael Reb, Is the List of Incomplete Open Cubes Complete?, author's preprint dated July 13, 2014 (math.ksu.edu); published in Nexus Network Journal 17(3), 913–925 (2015), doi:10.1007/s00004-015-0254-8. The published version was not read. Quoted briefly.
- Michael Allan Reb, Analysis of variations of Incomplete open cubes by Sol Lewitt, Master of Science report, Kansas State University, 2013 (K-REx). In copyright; cited, not quoted.
- Mikael Vejdemo-Johansson, Incomplete Open Platonic Solids, arXiv:2602.20425 (February 2026), CC BY 4.0. Its Figure 1 reproduces LeWitt's 1974 poster, which is LeWitt's work rather than the paper's; this page read the poster there and does not reproduce it.
- OEIS A393969, entered by Paolo Xausa, oeis.org, CC BY-SA 4.0.
- Rosalind Krauss, LeWitt in Progress, October 6 (1978), also in The Originality of the Avant-Garde and Other Modernist Myths (MIT Press, 1985), 244–258; quoted from the excerpt in Mariabruna Fabrizi, Irrational Thoughts Should be Followed Absolutely and Logically, SOCKS, June 15, 2016. Krauss's words are hers; quoted briefly.
- Rob Weychert, About Incomplete Open Cubes Revisited, robweychert.com, 22 September 2018. Quoted briefly.
- Alison Pitt, An Exploration of Incomplete Open Cubes, alisonpitt.com, September 10, 2023. Quoted briefly.
The drawings on this page are its own, made from the 12 edges of a cube; no image, tracing or layout of LeWitt's chart is reproduced, and the only cells described are the five of its 10-edge row. The chart, the photographs and the frames are LeWitt's work, and this page treats them as in copyright.