Checking the Construction

Every claim on Drawn Mathematics is checked by the method that fits it: a number is calculated again, a diagram is generated from its rule, a puzzle answer is searched or proved, and a historical claim is traced to a source.

Mathematical statements

Statements are compared with the original publication wherever it can be read, and with standard references. Anything that can be calculated is calculated again from its definition rather than copied from a table.

Copyrighted references such as Wolfram MathWorld serve only as a starting point for checking. Their text, figures and tables are not copied; the numbers in this site's tables are recalculated or traced back to the original source.

Conditions stay attached to the statements they limit. If a result holds only for convex polygons, or only at a single interior vertex, the sentence says so. When the site counts something itself and has not compared the count with a published table, the article says that too. A regularity noticed in a finite set of examples is reported as an observation about those examples, not as a theorem.

Diagrams and data

Every diagram is generated from the rule or coordinates stated in its caption, so the parameters can be reused to redraw it. A caption such as "Koch curve, 4 iterations" names exactly what is drawn.

Three layers are kept apart. The mathematical limit is the ideal object, such as a fractal after infinitely many steps. A drawing on the page is a finite approximation, and its caption states how many steps it shows. A natural example, such as a coastline, is described with the scale at which it was observed and the source of its data, because self-similarity in nature holds only over a limited range of scales.

A finite drawing is a demonstration, not a proof. A general result comes with its assumptions and a proof, or a pointer to where the proof was published.

Any figure in which a length, an angle, a count or a label matters is drawn from coordinates. When an article computes a table, the same data feeds the text, any summary card and the downloadable CSV file, so they cannot disagree.

Puzzles

Every puzzle answer is checked before publication, either by an exhaustive search or by a proof. The article states how the check works, so you can repeat it: which cases were searched, on what grid, and what counts as the same answer.

When a general result rests on reducing every case to the few that a search covers, the article credits the published proof that makes the reduction. The search demonstrates the count; the proof establishes it.

Paintings and perspective

Painting analyses use public-domain images only, and works still in copyright are not analysed. Each analysis runs the vanishing audit and states three things: the projection assumed, which edges are treated as parallel and why, and where distortion in the image could come from (the camera lens, a warped canvas or panel, later restoration).

How far the overlay lines miss a single vanishing point is a geometric fact. Why they miss, whether through a choice of composition, an intention or an error, is interpretation. A painter's intention is stated only when a published source supports it; otherwise the explanation is labelled as this site's interpretive hypothesis.

Historical and cultural claims

Who proved, invented or painted what, and when, is given with a source. So is any sentence that says first, only or all. Where sources disagree, the more original source is preferred and the disagreement is noted in the text. General encyclopedias serve as leads, never as cited sources.

Patterns are described by their geometry. A statement about which culture a pattern belongs to carries a citation, and the site does not pronounce on the religious meaning of a pattern.

Crease-check levels

Each crease pattern carries a label naming the highest level it has actually reached, in the wording below. The three levels are:

• Level 1: local flat-folding conditions pass (Maekawa and Kawasaki at every interior vertex; not a proof that the sheet folds flat)
• Level 2: proven to fold flat (layer order solved, or a published proof)
• Level 3: folding sequence checked step by step

Level 1 is a computation. At every interior vertex, the number of mountain folds and the number of valley folds must differ by 2 (Maekawa's theorem), and the angles taken alternately around the vertex must each add up to 180° (Kawasaki's theorem). Both conditions are necessary for folding flat. They do not guarantee that the whole sheet folds flat, or that its layers can be stacked without passing through each other (E. Demaine and M. Demaine, "Recent Results in Computational Origami", 2001, §3.1–3.2; Maekawa's theorem as stated in T. Hull, "The Combinatorics of Flat Folds: a Survey", 2002, Theorem 2.2, arXiv:1307.1065).

Level 2 needs evidence beyond the vertex check: a solved layer order kept on record, or a published proof. A traditional model can rest on a published proof. An original crease pattern needs its own layer-order solution, and no original crease pattern appears on the site below this level.

Level 3 means the folded state after each step has been recorded and checked in turn. An animation that collapses every crease at the same time does not count as a checked sequence. A folding simulator of that kind solves the shape numerically and lets the paper stretch slightly, so it produces an approximate folded shape to look at. It does not prove that a folded state is valid, and it cannot raise a pattern from Level 1 to Level 2.

Words such as "foldable" and "verified" are used only for the level a pattern has actually reached, and no animation is presented as a guarantee that a sheet will fold.

Current status: a folding-simulator test was run on 2026-10-26 on six crease patterns drawn for this site: the square, waterbomb and bird bases, a traditional crane, and two patterns that pass the local conditions everywhere but cannot fold flat. The simulator settled every one of them into a nearly flat shape, including the two that cannot fold flat, where layers pass through each other. That is why any simulator render on this site is labelled as an approximate rendering, never as proof that a pattern folds. The check the site itself applies to its crease patterns is the Level 1 computation of the local conditions; a Level 2 label appears only where a published proof is cited.

Images and permissions

Photographs and scans are checked one file at a time. The license of each file is read from its own source page, and a copy of that page is kept on record; the reputation of a museum or an uploader is not taken as evidence. A collection label such as "No known restrictions" is a lead, not proof. Each image is also compared with its description before use, because a file can carry the right license and still show the wrong object.

Public-domain status is stated with its basis, usually the creator's year of death and the jurisdiction, because "published before a certain year" does not hold in every country.

Redrawing is not a way around copyright. Crease patterns by modern origami designers, named modern illusions and puzzle illustrations, and artworks still in copyright are not traced or redrawn. Diagrams generated from a definition are designed independently of any protected image.

Scene illustrations are captioned "Generated scene illustration — not a photograph, historical image or researched reconstruction". Their details are not historical evidence, and no geometric point rests on one. How each kind of image is licensed is on Borrow these drawings.

Corrections

When an error is found, the article gets a dated note at its foot and a matching entry on Corrections and redraws. A mathematical error also means the calculation or search behind it is repeated and the diagram regenerated from the corrected rule. Every place that repeats the claim changes on the same day, and so does the revision date in Cite this page. To report an error, see the contact page.