--- slug: the-same-cube-turned passed: 2026-11-02 by: ccr-81332fb3-8nrbtu wave, one independent agent on oversight/claims-pass.md (pre-deploy; reader check on local copies with AW_DIR; every source-record entry read against its kept copy and live source) checked: 105 confirmed: 79 wrong: 0 unverifiable: 0 observed: 26 stale: 0 --- # Claims pass: The Same Cube, Turned Live page: not deployed yet; step 2 skipped. Reader check (pre-deploy, local copies): `AW_DIR= node verify-the-same-cube-turned.mjs`, run with the verifier copied alone into a second empty directory: exit 0, 422 passed, 0 failed (in the repository, with kept copies and frontmatter: 478 passed, 0 failed; `--mutate`: 16 checks fail on the mutant, as they should). The page prints no transcript of the verifier. Step 6 (one sentence-only edit at a time, plain run, ten edits): "45 pairs" to 46, "5 frames, not 4" to 6, "at most 121" to 120, the dek's "He found 122" to 123, "12 turns" to 13, "Paolo Xausa in 2026" to 2025, "Figure 15" to 14, "2001 Wadsworth" to 2002, "October in 1978" to 1979, and "2 423 206" inside the quotation to 2 423 207: every one exits 1, each failing its own sentence check (or the quotation check) and the residual sweep. Mutation sweep: the-same-cube-turned: 360 mutated, 360 caught, 0 survived (text 0, meta 0, year 0). `assay/sources.mjs the-same-cube-turned --online`: 32 FOUND, 43 UNREADABLE, 23 in the repository, 0 NOT FOUND, 2 DOI MATCH. The 39 UNREADABLE PDFs (R&R preprint, Vejdemo-Johansson) were fetched with curl and their pdftotext output is byte-identical to the kept copies (preprint sha256 de2121f7...0b82, arXiv PDF 53909e4a...95b8, as recorded); the 4 HTTP 403 (K-REx PDF and item record) answer 200 to curl and were read by hand. Every quotation (13) was matched character for character against the live source text, and 40 further source sentences the page relies on (dates, titles, figures, attributions) were found in the live sources. The chart reading was made again from the poster image (same hash as recorded) and R&R's Figures 15 and 16 were rendered from the PDF and read; the figures were recounted by a third, independent program. ## Claims - CONFIRMED | 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. | https://socks-studio.com/2016/06/15/irrational-thoughts-should-be-followed-absolutely-and-logically-sol-lewitts-variations-of-incomplete-open-cubes-1974/ | live: "In 1974, American artist Sol LeWitt created one of his major works ... “Variations of Incomplete Open Cubes”"; Weychert (live): "first exhibited ... in 1974". The rules clause is entry dek-rules - CONFIRMED | 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. | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | live PDF (sha256 de2121f7...0b82, same as kept): "the structure should be three-dimensional ... the structure should be connected ... identical, if one can be transformed into another by a space rotation (but not by a reflection)" - CONFIRMED | Sol LeWitt's Variations of Incomplete Open Cubes, shown at the John Weber Gallery in New York in 1974 | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "Variations of Incomplete Open Cubes is recognized as one of the major projects of the artist"; the poster itself is headed "VARIATIONS OF INCOMPLETE OPEN CUBES" (viewed in arXiv Fig. 1), so "of", not Weychert's and Vejdemo-Johansson's "on", is the chart's own form - CONFIRMED | shown at the John Weber Gallery in New York in 1974 | https://arxiv.org/html/2602.20425v1 | live caption: "Figure 1: Exhibition poster for Sol LeWitt’s Variations on Incomplete Open Cubes from a show at the John Weber Gallery in 1974." - CONFIRMED | shown at the John Weber Gallery in New York in 1974 | https://v6.robweychert.com/blog/2018/09/about-cubes-revisited | live: "Sol LeWitt first exhibited his Variations on Incomplete Open Cubes at the John Weber Gallery in New York City in 1974." R&R ref. [4] also: "The John Weber Gallery, New York, 1974" - CONFIRMED | an installation of 122 frames, each a cube with edges taken away, and a chart of drawings | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "It is represented by an installation of 122 frame structures, that is supplemented by drawings and diagrams of the structures." - CONFIRMED | a chart of drawings that labels them by how many edges they keep | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "He organized the list of incomplete cubes in groups according to the number of edges of the structures." The poster labels confirm it (entry labels) - OBSERVED | from 3/1 to 11/1 | recreated: curl -sSLO https://arxiv.org/html/2602.20425v1/le-witt-incomplete-open-cubes-01.jpg (sha256 8e95fa84...4fee9, the hash the kept reading records), viewed whole and with the 10-edge row cropped and enlarged three times | the first cell is labelled 3/1 and the last 11/1; every label is edges/index - CONFIRMED | He found 122. | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | Abstract: "the artist found the correct number of structures (that is 122)" - CONFIRMED | 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. | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "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 ... they correspond to variations 10/4 and 10/5 in Sol LeWitt’s list" - CONFIRMED | Natasha Rozhkovskaya and Michael Reb | https://api.crossref.org/works/10.1007/s00004-015-0254-8 | live Crossref: given "Natasha" family "Rozhkovskaya" (first), given "Michael" family "Reb" - OBSERVED | This page found no statement of the rules in LeWitt's own words. | research/the-same-cube-turned/README.md | the README records the search ("No statement of the rules in LeWitt's own words was found", with what was searched); my reading of all kept sources found none either. A statement about this page's search, worded as such - OBSERVED | 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. | research/the-same-cube-turned/README.md | checked in the sources themselves: R&R and Reb quote LeWitt only on why he asked no mathematician ([5] p.25); Vejdemo-Johansson quotes him only on the cube as "the best form to use as a basic unit" (1966). None quotes him stating the rules - CONFIRMED | Natasha Rozhkovskaya and Michael Reb in a preprint of 2014 | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "The rules followed by LeWitt for production of Incomplete open cubes are quite simple." then the three rules - CONFIRMED | in a preprint of 2014 (published in 2015) | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | title page "July 13, 2014" (PDF CreationDate also 2014-07-13); published 2015 is entry rr-published - CONFIRMED | in a preprint of 2014 (published in 2015) | https://api.crossref.org/works/10.1007/s00004-015-0254-8 | "published-print":{"date-parts":[[2015,12]]}; published-online 2015-07-16 - CONFIRMED | published in Nexus Network Journal 17(3), 913–925 (2015) | https://api.crossref.org/works/10.1007/s00004-015-0254-8 | "container-title":["Nexus Network Journal"] - CONFIRMED | Nexus Network Journal 17(3), 913–925 (2015) | https://api.crossref.org/works/10.1007/s00004-015-0254-8 | "page":"913-925" - CONFIRMED | Nexus Network Journal 17(3) | https://api.crossref.org/works/10.1007/s00004-015-0254-8 | "issue":"3", "volume":"17" - CONFIRMED | Reb in his Master of Science report of 2013 | https://krex.k-state.edu/server/api/core/bitstreams/dad3d273-0a8f-4d29-acf2-77cb6e715006/content | read by hand (the tool's HTTP 403 is its client; curl gets 200 and the PDF's text equals the kept copy): Abstract "three-dimensional and connected ... identical if one can be transformed into another by a space rotation (but not reflection)" - CONFIRMED | Michael Allan Reb, Analysis of variations of Incomplete open cubes by Sol Lewitt, Master of Science report, Kansas State University, 2013 | https://krex.k-state.edu/server/api/core/bitstreams/dad3d273-0a8f-4d29-acf2-77cb6e715006/content | title page: "MICHAEL ALLAN REB ... MASTER OF SCIENCE ... KANSAS STATE UNIVERSITY Manhattan, Kansas 2013"; K-REx dc.date.issued 2013-05-14 - CONFIRMED | and Mikael Vejdemo-Johansson in 2026 | https://arxiv.org/pdf/2602.20425 | section 2: "Sol LeWitt defines an incomplete open cube as a connected and non-planar subset of the edges of the cube and enumerates them up to rotational equivalence"; arXiv 23 Feb 2026 - CONFIRMED | the structure should be connected. | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | verbatim, character for character, against the live PDF text - CONFIRMED | a square, or just one edge are not listed in the project, since they are not three-dimensional. | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | verbatim against the live PDF text (the source sentence begins "For example,", which the page omits without changing the sense) - CONFIRMED | two structures are considered to be identical, if one can be transformed into another by a space rotation (but not by a reflection). | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | verbatim against the live PDF text - CONFIRMED | Vejdemo-Johansson asks for a connected and non-planar proper and non-empty subset of the edges | https://arxiv.org/pdf/2602.20425 | verbatim: "a connected and non-planar proper and non-empty subset of the set of edges of the polyhedron" - CONFIRMED | An incomplete cube lacks at least one edge. | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "the graphs in Figure 13 are not present in the Sol LeWitt’s Incomplete Open Cubes list, since ... the last one is the complete cube"; Vejdemo-Johansson: "proper ... subset" - OBSERVED | LeWitt's chart has 122, and every row matches. | recreated: curl -sSLO https://arxiv.org/html/2602.20425v1/le-witt-incomplete-open-cubes-01.jpg (sha256 8e95fa84...4fee9, the hash the kept reading records), viewed whole and with the 10-edge row cropped and enlarged three times | rows counted on the poster: 3 + 5 + 14 + 24 + 32 + 25 + 13 + 5 + 1 = 122; recount.py gives the same nine tallies for the default rules - OBSERVED | 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. | recreated: curl -sSLO https://arxiv.org/html/2602.20425v1/le-witt-incomplete-open-cubes-01.jpg (sha256 8e95fa84...4fee9, the hash the kept reading records), viewed whole and with the 10-edge row cropped and enlarged three times | 3/1-3/3; 4/1-4/5; 5/1-5/14; 6/1-6/14 and 6/15-6/24; 7/1-7/14, 7/15-7/28, 7/29-7/32; 8/1-8/14 and 8/15-8/25; 9/1-9/13; 10/1-10/5; 11/1, as the page says - OBSERVED | Of the 24 ways to set the four switches, only one reproduces those rows, and it is the setting you started with. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | third independent count (orbits of frozensets, no bitmasks, no Burnside): 24 settings, exactly one matches the chart, the default - OBSERVED | 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. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | mirror-as-same gives 2, 3, 9, 14, 19, 16, 9, 4, 1 = 77; only the rotation-only settings can give the chart's rows - CONFIRMED | The same nine numbers are sequence A393969 in the On-Line Encyclopedia of Integer Sequences, entered by Paolo Xausa in 2026. | https://oeis.org/search?q=id:A393969&fmt=text | live: "%S A393969 3,5,14,24,32,25,13,5,1" and "%A A393969 _Paolo Xausa_, Mar 31 2026" (the entry misspells "Sol DeWitt"; the page does not copy it) - OBSERVED | With 3 edges there are 3 frames: three edges meeting at one corner, and two staircases that each take one step in every direction. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | 3 frames with 3 edges, 1 a corner of three; the other two are mirror images in different rotation classes. The poster's 3/1-3/3 show the same three shapes - OBSERVED | Across the 122 frames the rules allow, 45 pairs are mirror twins and 32 frames are their own mirror image | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | 32 self-mirror classes, 45 pairs; 2 x 45 + 32 = 122, 45 + 32 = 77 - CONFIRMED | A computer does all of this in a blink | https://arxiv.org/pdf/2602.20425 | section 3.3: "cube (0.079 seconds)" on a 32-core server; the page's own recount ran in under a second here - CONFIRMED | which recalls an old contrast in writing about the work | https://socks-studio.com/2016/06/15/irrational-thoughts-should-be-followed-absolutely-and-logically-sol-lewitts-variations-of-incomplete-open-cubes-1974/ | Krauss excerpt (live): "[U]nlike the algebraic expression of the expansion of a given series ... The babble of a LeWitt serial expansion has nothing of the economy of the mathematician’s language." It is about Variations of Incomplete Open Cubes by name - CONFIRMED | Rosalind Krauss, in her essay LeWitt in Progress (first published in the journal October in 1978), set his serial lists against algebra | https://socks-studio.com/2016/06/15/irrational-thoughts-should-be-followed-absolutely-and-logically-sol-lewitts-variations-of-incomplete-open-cubes-1974/ | SOCKS names the essay "LeWitt in Progress" and its excerpt sets LeWitt's series against "the algebraic expression of the expansion of a given series"; October 1978 per Crossref (entries krauss-1978, krauss-issued) - MINOR | first published in the journal October in 1978 | https://api.crossref.org/works/10.2307/778617 | Crossref: "LeWitt in Progress", Krauss, "container-title":["October"], volume 6, 1978, page 46: October and 1978 are supported. "first" is true (Hyperallergic, https://hyperallergic.com/delirious-art-at-the-limits-of-reason-met-breuer-2017.md: "“LeWitt in Progress,” published in the Autumn 1978 issue of October magazine"; JSTOR 778617: October, Vol. 6 (Autumn, 1978), pp. 46-60) but no excerpt in the record says it: see Findings - CONFIRMED | (first published in the journal October in 1978) | https://api.crossref.org/works/10.2307/778617 | "issued":{"date-parts":[[1978]]} - CONFIRMED | The babble of a LeWitt serial expansion has nothing of the economy of the mathematician’s language. | https://socks-studio.com/2016/06/15/irrational-thoughts-should-be-followed-absolutely-and-logically-sol-lewitts-variations-of-incomplete-open-cubes-1974/ | verbatim against the live SOCKS page, curly apostrophe included; the page says it quotes the SOCKS excerpt, not the journal - CONFIRMED | The words are quoted from the excerpt of the essay on SOCKS. | https://socks-studio.com/2016/06/15/irrational-thoughts-should-be-followed-absolutely-and-logically-sol-lewitts-variations-of-incomplete-open-cubes-1974/ | "Here’s an excerpt:" after the citation of "LeWitt in Progress" - CONFIRMED | Burnside's lemma counts the frames without listing a single one. | https://arxiv.org/pdf/2602.20425 | section 1: "using Burnside’s Lemma to validate Sol LeWitt’s count [2]"; the mathematics recreated: the identity, quarter, face-half, third and edge-half rows sum to 2,928 and 2,928 / 24 = 122 without forming any class (recount.py) - OBSERVED | 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. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | identity 1 x 2,366; quarter 6 x 2 = 12; face half 3 x 22 = 66; third 8 x 11 = 88; edge half 6 x 66 = 396; 2,928 / 24 = 122. Every rotation of a kind fixes the same number - OBSERVED | A skew pair comes in two hands, mirror images of each other, so the row has 5 frames, not 4. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | 10-edge classes by missing pair: corner, face, opposite, skew, skew - OBSERVED | 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 | recreated: curl -sSLO https://arxiv.org/html/2602.20425v1/le-witt-incomplete-open-cubes-01.jpg (sha256 8e95fa84...4fee9, the hash the kept reading records), viewed whole and with the 10-edge row cropped and enlarged three times | my own reading of the enlarged crop: 10/1 lacks AB, OF; 10/2 OC, OF; 10/3 OB, OF; 10/4 CD, OF; 10/5 OA, OF: the same as lewitt-chart-reading.txt - OBSERVED | The diagrams of 10/4 and 10/5 that Rozhkovskaya and Reb print in their Figure 15, as LeWitt presented them, read the same. | recreated: pdftoppm -r 300 -f 9 -l 9 -png /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/dl/rr.pdf, then viewed Figure 15 | R&R's line diagrams: 10/4 lacks the lower-right side CD and the spoke OF; 10/5 lacks the spokes OA and OF; the same as the poster. Their Figure 16 corrected 10/5 lacks AB and EF - CONFIRMED | that Rozhkovskaya and Reb print in their Figure 15, as LeWitt presented them | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "Here are the diagrams of these two embeddings as they were presented by the artist:" - CONFIRMED | their Figure 15 | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "Figure 15: Incomplete cubes 10/4 and 10/5 in the original Sol LeWitt’s list." - CONFIRMED | 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 | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | verbatim against the live PDF text (Abstract) - CONFIRMED | 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 | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | verbatim against the live PDF text - CONFIRMED | at 10/4 and 10/5 | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "they correspond to variations 10/4 and 10/5 in Sol LeWitt’s list" - OBSERVED | 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. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/row10.py | rotations carrying 10/5 onto 10/4: one third turn and one edge-axis half turn, under the page's reading and under the depth-reversed one - CONFIRMED | The other hand of the skew pair, the frame that belongs in one of those two cells, is not drawn. | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "the correct version, where one of the two structures in the original list is substituted by its mirror image"; recreated: the mirror of 10/4 is in none of the five cells (row10.py) - OBSERVED | So the chart has 122 cells but shows at most 121 different cubes, and the number 122 is still right. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/row10.py | 4 distinct frames in the 5 cells, so 122 cells show at most 121 frames - CONFIRMED | Rozhkovskaya and Reb report a mistake in one pair | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "his list contains a mistake in the presentation of a pair of incomplete cubes" - OBSERVED | It has not checked the other 117 drawings one by one. | research/the-same-cube-turned/sources/lewitt-chart-reading.txt | section 4: "the other 117 cells were not read cell by cell for this page"; 122 - 5 = 117 - OBSERVED | 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. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/row10.py | the poster's cells are isometric hexagons whose centre stands for two corners; the depth-reversed labelling is the mirror image of the first and every conclusion holds under both - CONFIRMED | Rozhkovskaya and Reb write that LeWitt made small three-dimensional models to check that no frame was repeated | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "In order to check that there are no repetitions, the artist made small three-dimensional models." - CONFIRMED | citing the 2001 Wadsworth Atheneum catalogue (which this page did not read) | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | ref. [5] "Sol LeWitt, Incomplete Open Cubes, Wadsworth Atheneum Museum of Art, Hartford, Connecticut, 2001."; the claim is cited as "([5], p.25)" - CONFIRMED | that he later asked two mathematicians to check the list | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "According to ([5], p.25), LeWitt eventually asked professional mathematicians to check his list. That was done by Dr. Erna Herrey and by Professor Arthur Babakhanian": two - CONFIRMED | who confirmed to the artist that the number of structures is 122 | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | verbatim against the live PDF text - CONFIRMED | Rob Weychert's Incomplete Open Cubes Revisited (2018) | https://v6.robweychert.com/blog/2018/09/about-cubes-revisited | live: "Exploring the extended potential of Sol LeWitt’s ‘Incomplete Open Cubes’ 22 Sep 2018" - CONFIRMED | has found and rendered a total of 4,094 variations (including LeWitt’s original 122) | https://v6.robweychert.com/blog/2018/09/about-cubes-revisited | verbatim against the live page, curly apostrophe included - CONFIRMED | That is every subset of the 12 edges except none and all, 4,096 − 2, with no turning: a different question, not a correction. | https://arxiv.org/pdf/2602.20425 | "excluding the empty cube and the full cube, the remaining cubes total a count of 4094 if one includes both planar, disconnected and rotated"; Weychert counts rotated forms as unique; 4,096 - 2 = 4,094 - CONFIRMED | Weychert describes the cubes this way: each part is connected to at least one other part | https://v6.robweychert.com/blog/2018/09/about-cubes-revisited | verbatim: "All of LeWitt’s cubes are contiguous; each part is connected to at least one other part." - OBSERVED | 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. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | 135 classes under touching, 13 of them disconnected, every one in exactly 2 pieces - CONFIRMED | Alison Pitt's 2023 re-enumeration states the test as | https://www.alisonpitt.com/blog/an-exploration-of-incomplete-open-cubes | live, dated "September 10, 2023": "Contiguity: does each edge touch at least one other edge?" (she takes the criteria from Weychert, and states them) - CONFIRMED | does each edge touch at least one other edge? | https://www.alisonpitt.com/blog/an-exploration-of-incomplete-open-cubes | verbatim against the live page - CONFIRMED | then traces whole paths through each frame | https://www.alisonpitt.com/blog/an-exploration-of-incomplete-open-cubes | "traverse every possible contiguous path from every possible starting edge, and if the set of contiguous edges described by the paths matched the overall configuration, then the configuration was valid" - CONFIRMED | I was left with exactly 122 configurations, the same as LeWitt’s. | https://www.alisonpitt.com/blog/an-exploration-of-incomplete-open-cubes | verbatim against the live page, curly apostrophe included - CONFIRMED | Vejdemo-Johansson reads three-dimensional as non-planar (the corners do not all lie in one plane) | https://arxiv.org/pdf/2602.20425 | section 2: "connected and non-planar"; where R&R and Reb write "three-dimensional", he writes "non-planar" - CONFIRMED | (the corners do not all lie in one plane) | https://arxiv.org/pdf/2602.20425 | section 3.3 tests planarity by whether each further vertex lies in the plane through the first three - OBSERVED | For frames in one piece the two readings never disagree, so both give 122. | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | 0 of the connected subsets where "uses all three directions" and "corners not coplanar" disagree; not-flat gives 122, 150 (touching), 210 (no rule) - CONFIRMED | Vejdemo-Johansson applied the same rules to all five Platonic solids | https://arxiv.org/pdf/2602.20425 | Abstract: "enumerating all the incomplete open platonic solids"; section 2 defines one rule set for every polyhedron, counted up to rotation - CONFIRMED | and reports 6 tetrahedra, 122 cubes (just like LeWitt), 185 octahedra, 2 423 206 dodecahedra and 16 096 166 icosahedra | https://arxiv.org/pdf/2602.20425 | verbatim (Abstract); the page prints the PDF's thin spaces as no-break spaces - OBSERVED | 6 tetrahedra (6 edges, 12 turns) and 185 octahedra (12 edges, 24 turns), as he reports | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/solids.py | tetrahedron 6 edges, 12 turns, 6 classes; octahedron 12 edges, 24 turns, 185 classes - CONFIRMED | The dodecahedron and the icosahedron have 30 edges each, so 1,073,741,824 subsets each | https://arxiv.org/pdf/2602.20425 | footnote 2: "dodecahedron and icosahedron: 30" - CONFIRMED | so 1,073,741,824 subsets each | https://arxiv.org/pdf/2602.20425 | "1 073 741 824 subsets in total" (2^30, per solid; the page's "each" is the arithmetic) - OBSERVED | 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. | recreated: curl -sSLO https://arxiv.org/html/2602.20425v1/le-witt-incomplete-open-cubes-01.jpg (sha256 8e95fa84...4fee9, the hash the kept reading records), viewed whole and with the 10-edge row cropped and enlarged three times | the image is Figure 1 of arXiv:2602.20425v1; its hash matches the one the reading records - CONFIRMED | Figure 1 of Vejdemo-Johansson's paper | https://arxiv.org/html/2602.20425v1 | live caption of Figure 1 as quoted - MINOR | Natasha Rozhkovskaya and Michael Reb, Is the List of Incomplete Open Cubes Complete?, author's preprint dated July 13, 2014 | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | title and "July 13, 2014" are on the PDF; the PDF carries no author line. The authorship is true (it sits in Rozhkovskaya's KSU directory, and https://www.math.ksu.edu/~rozhkovs/artessays.html lists the paper with the same abstract under "Authors: M. Reb, N. Rozhkovskaya"), but no excerpt in the record says so: see Findings - CONFIRMED | 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. | https://api.crossref.org/works/10.1007/s00004-015-0254-8 | title, volume 17, issue 3, pages 913-925, 2015; DOI MATCH from the online check - CONFIRMED | In copyright; cited, not quoted. | https://krex.k-state.edu/server/api/core/items/49e20a9a-5b16-430e-8ea6-3ef33381bc0e | read by hand (curl 200): dc.rights "© the author. This Item is protected by copyright and/or related rights." - CONFIRMED | Master of Science report, Kansas State University, 2013 (K-REx) | https://krex.k-state.edu/server/api/core/items/49e20a9a-5b16-430e-8ea6-3ef33381bc0e | dc.description.degree "Master of Science"; dc.title "Analysis of variations of Incomplete open cubes by Sol Lewitt"; the page's handle http://hdl.handle.net/2097/15809 resolves to this item - CONFIRMED | Mikael Vejdemo-Johansson, Incomplete Open Platonic Solids, arXiv:2602.20425 (February 2026), CC BY 4.0. | https://arxiv.org/pdf/2602.20425 | "Incomplete Open Platonic Solids Mikael Vejdemo-Johansson ... arXiv:2602.20425v1 [math.CO] 23 Feb 2026" - CONFIRMED | arXiv:2602.20425 (February 2026), CC BY 4.0 | https://arxiv.org/abs/2602.20425 | live abstract page and OAI record: http://creativecommons.org/licenses/by/4.0/ - CONFIRMED | OEIS A393969, entered by Paolo Xausa, oeis.org, CC BY-SA 4.0. | https://oeis.org/wiki/The_OEIS_End-User_License_Agreement | live: "Creative Commons Attribution Share-Alike 4.0 license (CC-BY-SA-4.0)"; author Paolo Xausa per %A - CONFIRMED | Rosalind Krauss, LeWitt in Progress, October 6 (1978) | https://api.crossref.org/works/10.2307/778617 | title "LeWitt in Progress", volume "6", author Rosalind Krauss, 1978; DOI MATCH - MINOR | Rosalind Krauss, LeWitt in Progress, October 6 (1978), reprinted in The Originality of the Avant-Garde and Other Modernist Myths (MIT Press, 1985), 244–258 | https://socks-studio.com/2016/06/15/irrational-thoughts-should-be-followed-absolutely-and-logically-sol-lewitts-variations-of-incomplete-open-cubes-1974/ | SOCKS gives "The Originality of the Avant-Garde and Other Modernist Myths (Cambridge and London: MIT Press, 1985): 244-258" and Crossref gives October 6 (1978); that the 1985 text reprints the 1978 essay is true (same title and author; Hyperallergic quotes the October essay's aporia sentence, which is also the last sentence of the SOCKS excerpt from the 1985 book) but no excerpt in the record says it: see Findings - CONFIRMED | Mariabruna Fabrizi, Irrational Thoughts Should be Followed Absolutely and Logically, SOCKS, June 15, 2016 | https://socks-studio.com/2016/06/15/irrational-thoughts-should-be-followed-absolutely-and-logically-sol-lewitts-variations-of-incomplete-open-cubes-1974/ | "June 15, 2016 by Mariabruna Fabrizi"; title shortened from “Irrational Thoughts Should be Followed Absolutely and Logically”: Sol LeWitt’s “Variations of Incomplete Open Cubes” (1974) - CONFIRMED | Rob Weychert, About Incomplete Open Cubes Revisited, robweychert.com, 22 September 2018. | https://v6.robweychert.com/blog/2018/09/about-cubes-revisited | "About Incomplete Open Cubes Revisited ... 22 Sep 2018" - CONFIRMED | Alison Pitt, An Exploration of Incomplete Open Cubes, alisonpitt.com, September 10, 2023. | https://www.alisonpitt.com/blog/an-exploration-of-incomplete-open-cubes | "An Exploration of Incomplete Open Cubes Alison Pitt - Admin September 10, 2023" - CONFIRMED | In 1974 Sol LeWitt tried to draw every different way to do it, counting a shape turned around as the same shape, and he counted 122, which is right. | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | Abstract "(that is 122)"; rotation-same per the rules; "In 1974" per dek-1974 - CONFIRMED | But two of his drawings are one shape turned around, and the mirror image of that shape is missing. | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "put the same incomplete open cube twice (the second time rotated), instead of two structures that are mirror images of each other" - CONFIRMED | Classifying the whole list, they confirmed the count | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | "This approach allowed us to confirm that there are indeed 122 different incomplete open cubes – LeWitt found all incomplete open cubes." - CONFIRMED | The chart, the photographs and the frames are LeWitt's work | https://v6.robweychert.com/blog/2018/09/about-cubes-revisited | "122 2⅝ʺ wooden cube frames, each with various pieces missing, as well as schematic drawings and photographs of the same" - OBSERVED | 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 | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | the empty subset is counted in exactly the 4 settings with one piece and three-dimensional both off - OBSERVED | 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) | recreated: node verify-the-same-cube-turned.mjs (in the repository) | 478 passed, 0 failed, including "F: quotes.json [k] is in " for all 11 passages; separately all 13 quotations matched the live sources (pageq.py) - CONFIRMED | (added) Its Figure 1 reproduces LeWitt's 1974 poster, which is LeWitt's work rather than the paper's | https://arxiv.org/html/2602.20425v1 | caption "Exhibition poster for Sol LeWitt’s Variations on Incomplete Open Cubes from a show at the John Weber Gallery in 1974"; the image is the chart headed VARIATIONS OF INCOMPLETE OPEN CUBES - CONFIRMED | (added) The mathematicians who have studied his list state them the same way | https://www.math.ksu.edu/~rozhkovs/LeWitt_cubes.pdf | R&R (three-dimensional, connected, rotation not reflection), Reb (the same words, K-REx PDF) and Vejdemo-Johansson (connected, non-planar, rotation, mirror images allowed) as named after the colon; OEIS A393969 %C states the same three. The page itself names the non-planar reading as a second reading - OBSERVED | (added) the other computed figures: 4,096 ways to choose; 45 + 45 + 32 = 122 and 45 + 32 = 77; 150 instead of 135 and 210 instead of 175; 2,366; 122, 77, 135, 175, 126, 123 and 218 among the 24 settings; the four kinds of missing pair; that weak touching is true of every listed frame | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/recount.py | all reproduce in an implementation that shares no code with engine.mjs or the verifier (frozensets of corner pairs, orbit flooding, determinant test for flatness); arithmetic redone by hand - CONFIRMED | (added) 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; Run it yourself (Node 18 or later, nothing to install) | verify-the-same-cube-turned.mjs | lines 72-90 (corners renumbered, closure of rx, ry and mz), 101-106 (union-find), 133-146 (Burnside over cycles); imports only node: builtins; the reader run from an empty directory passed under Node 22 - OBSERVED | (added) 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 | public/strata/the-same-cube-turned/app.mjs | index.html has no img element and no image source; app.mjs draws every frame as SVG lines from a 12-bit mask in its own projection (yaw -28, pitch 23 degrees, not isometric) and its own order; the only cells named are 10/1 to 10/5 - OBSERVED | (added) the sentences app.mjs writes as the reader acts: "A third of a turn about a long diagonal carries 10/5 exactly onto 10/4 ... (A half turn about the line through two edge midpoints does it too.)"; "Mirrored, 10/4 becomes a frame no turn can reach ... None of the five cells in his row is this frame"; "of the 13 columns differ"; the ten names of the kinds of symmetry | recreated: python3 -I /tmp/claude-0/-home-user-Artificial-Wasteland/bb727bf8-61a8-5b0c-9312-348d724897c9/scratchpad/claims/the-same-cube-turned/scripts/row10.py | row10.py finds exactly those two turns and no cell holding the mirror image; the tally has 13 columns (0 to 12); the ten names checked geometrically: -(quarter turn) is a quarter-turn rotoreflection, -(third turn) a sixth-turn rotoreflection, -(face half turn) a mirror parallel to two faces, -(edge half turn) a mirror through two opposite edges, and the verifier's section G rebuilds each from its words - CONFIRMED | (added) the frontmatter relates notes: the-count-that-ran-off-the-page held a cube count against a published record that ran out; a mirror reverses one axis (mirror-reversal); eleven-moves-from-anywhere holds one corner of a pocket cube still | src/content/strata/the-same-cube-turned.md | all three slugs exist; their pages say "the only published record was a five-term comment in OEIS A228267", "A plane mirror reverses one axis only", and "fixing the down-back-left corner quotients out whole-cube rotation" ## Findings - [fixed] MINOR (claim lines "first published in the journal October in 1978" and the Krauss citation "reprinted in The Originality ..."): no kept source says "first" or "reprinted". Re-read: Crossref 10.2307/778617 (kept crossref-records-json.txt) gives "container-title":["October"] and "issued":{"date-parts":[[1978]]}; SOCKS (kept, and live) gives "Rosalind E. Krauss, “LeWitt in Progress,” The Originality of the Avant-Garde and Other Modernist Myths (Cambridge and London: MIT Press, 1985): 244-258". Searched for words saying either, 2026-11-02: Hyperallergic says only "“LeWitt in Progress,” published in the Autumn 1978 issue of October magazine"; the CCA library record of the book says "fifteen essays, most of which originally appeared in October" (not this essay by name); the UC Irvine Krauss bibliography failed TLS verification here (self-signed certificate in chain; not bypassed), the Wayback Machine was unreachable, the Internet Archive's copy of the book (originalityofava0000krau) is restricted and its search-inside refused, and JSTOR 778617 served a bot challenge. So the page now says only what the kept words say: "(published in the journal October in 1978)" (public/strata/the-same-cube-turned/index.html:206) and "October 6 (1978), also in The Originality of the Avant-Garde and Other Modernist Myths (MIT Press, 1985), 244–258" (index.html:311). The same change in quotes.json (rebuilt by research/the-same-cube-turned/make-quotes.mjs: "October 6, 1978; also in a 1985 collection, pp. 244-258"), the verifier's CITES entry s-krauss, its says() needle (now "in her essay LeWitt in Progress (published in the journal October in 1978)", so a reinserted "first" fails it; tested on a scratch copy: 2 failed) and its FACTS label, the kept SOCKS copy's own header line, the README row, and source-record entries krauss-essay, krauss-1978, krauss-issued and c-krauss (claims now word for word on the page) and old-contrast's where. Planting "reprinted in" back in a scratch copy fails the s-krauss citation check. - [fixed] MINOR (entry c-rr: the preprint PDF names no author): re-read the kept PDF text, which has title and date only. Fetched https://www.math.ksu.edu/~rozhkovs/artessays.html (2026-11-02, HTTP 200, HTML sha256 6b8ffa02...4f45), Rozhkovskaya's page in the same directory as the PDF: "Is the List of Incomplete Open Cubes Complete? Authors: M. Reb, N. Rozhkovskaya Info: Final version published in Nexus Network Journal, vol. 17, no. 3 (2015)." Its abstract's three sentences are word for word in the PDF's abstract, which has one sentence more (so "the same abstract" would overstate it); her Publications.html also lists "Reb.M, Rozhkovskaya, N., Is The List of Incomplete Open Cubes Complete?". Kept as research/the-same-cube-turned/sources/rozhkovskaya-artessays.txt (the heading and that entry only), added to the source record as entry c-rr-authors beside c-rr (assay/sources.mjs --online: FOUND), bound in the verifier's FACTS (a scratch copy with the authors line cut fails it), and a README row added. No page change. After both fixes: verifier 479 passed, 0 failed; --mutate red (16 failed); check-sources 99 entries, sound; online 33 FOUND, 43 UNREADABLE, 23 in the repository, 0 NOT FOUND; mutation sweep 360 mutated, 360 caught, 0 survived.