Which Layer Was Checked

A diagram can look right and still be wrong. A knot drawn with its crossings flipped is a different knot, and a pop-up piece can clear the page when the book is open yet jam halfway there. So every diagram on Fold, Knot & Bind carries a caption that names the checks it passed, and it claims nothing beyond them.

Knot diagrams

Each knot is drawn from a centerline, the path through the middle of the rope. At every crossing, which strand passes over and which passes under is computed from that path, not drawn by eye. A knot diagram counts as checked only when all three of these hold:

1. The crossings are computed from the drawn centerline.
2. With its ends joined into a closed loop, the drawing gives the expected knot type, judged by knot invariants: values such as the Jones polynomial that stay the same however a closed loop is pushed around.
3. The crossings match a plate in a public-domain manual, such as Verrill (1912 or 1917), Dana (1847) or Lever (1843), one crossing at a time. The caption names the printing and figure that were compared.

Matching the written instructions is not enough. An early drawing of the clove hitch looked right and agreed with the written method crossing by crossing, yet with its ends joined it computed as the wrong knot type. The invariants caught it before the drawing was used.

Some knots, the bowline among them, have no closed form that a published fact can confirm. For these, every crossing must match the written method, and the page names no knot type for the closed loop. Until the plate comparison is done, a caption states only what was computed, for example "crossings computed from the drawn centerline; closure invariants match the expected knot type." Some knots and steps appear in no public-domain plate; those drawings are used once every crossing matches the written method, and their captions name only the checks that were done. Easily confused knots, such as the square and granny knots, are drawn side by side. Step-by-step diagrams appear only for simple knots, and each step must continue the crossings of the one before.

Paper mechanisms

Pop-ups and moving paper are drawn from a rigid-sheet model, which this site calls paper logic: each piece of paper is a flat sheet with no thickness that turns only along its creases. Captions say that paper thickness is not modeled.

A pop-up moved by the pages themselves, such as a v-fold or a parallel fold, is followed from shut to open and shut again in steps of 2 degrees. At every step, each piece must stay rigid and turn only at its creases, with no new crease; no piece may pass through another piece or a page; the turning pages alone must set every piece's position; and when the book is shut, every piece must lie inside the page edges. These conditions follow the properties of a pop-up listed by Li, Ju, Gu and Hu (2011).

Pull tabs, turning paper wheels called volvelles, lift flaps and tunnel books form a second group, because the reader pulls, turns, lifts or draws them out and the pages alone do not move them. This grouping is the site's own. These are checked for travel (the full pull or turn and where it stops), collision between layers, rivets and slots, storage inside the page at every sampled setting (every 2° step for a volvelle), and linkage, meaning each moving part's position follows from how far the control has moved. Where a reading can fall exactly on a dividing line, the page says so. Where parts are fixed only at the two ends of their movement, as tunnel-book panels are, only those end positions are drawn.

In the folding models, meaning the pop-ups moved by the pages and folds such as the Miura fold, collisions are checked in two ways: panels are tested for overlap at every step, and every fold angle is tracked for a sudden jump. The second test exists because a panel can pass through another at the instant both lie flat, when neither neighboring step shows any overlap. The hand-operated pieces in the group above are checked for collision between their layers instead.

The pages also say where the model and real paper part company. As a v-fold closes, the gap between each folded piece and the facing page shrinks with the opening angle; measured 5 mm out from the spine corner where the sheets meet, the narrower of the two pieces' gaps in the model is 0.07 mm at a 2° opening and 0.36 mm at 10°. That is a number from a model with no paper thickness, and at that point the model no longer describes real paper.

Crease patterns

Crease patterns appear only for folds that work as mechanisms, such as the Miura fold, and are drawn from the fold's parameters, not traced from a published drawing. Two levels are reported: Level 1, the standard local conditions for folding flat around each interior vertex (Maekawa's and Kawasaki's) hold; Level 2, the mountain and valley folds as marked meet them. Higher levels are claimed only where computed. The Miura pattern reaches Levels 1 and 2, and its rigid folding motion was sampled to 99.5% of fully folded with no collisions. The layer order of the fully folded form was not computed, so no page claims the pattern presses completely flat.

Loops, stitches and bindings

Knitting and crochet diagrams are drawn from settings that fix which way each loop passes through the next, compared with the descriptions in Thérèse de Dillmont's Encyclopedia of Needlework, and captioned as schematics of loop structure, not to scale. Hitomezashi grids are checked for an even number of stitch ends at every grid point and for a two-color split of the pattern. Celtic knotwork is checked for strict over-and-under alternation, and its separate cords are counted. A pattern count is computed and compared with a published count before it appears; if they disagree, no number is given. Bookbinding cross-sections are schematics after Roberts and Etherington's dictionary, not checked geometrically and not to scale. Thread routes for stab binding follow the National Diet Library handout on four-hole binding and are checked by a route calculation: each pair of neighboring holes is joined once on each face, and each hole is passed at least three times. Page-order drawings of nested folded sheets, such as a booklet's page layout, use no paper thickness. Push-out figures assume a paper thickness that the caption or table states, and drawings of nested sheets show the gaps far wider than real paper; the results are model numbers, not measurements.

Sources

When sources disagree, first-hand documents come first: the original manual, the paper that states a result, or a collection's record of the object. An institution's general description comes next, and encyclopedic summaries last. General encyclopedias and wikis are used only to find leads and are never cited. Sources of the same rank that disagree are set side by side.

Dates, failure and strength data, definitions of terms, origin stories and any claim to be first or earliest always carry a source. The earliest recorded example and the first invention are kept apart. When no readable source records where a pattern began, the page says the origin is unrecorded.

Bookbinding terms follow Roberts and Etherington, Bookbinding and the Conservation of Books: A Dictionary of Descriptive Terminology (Library of Congress, 1982), named on each page that uses them. Knot names and figure numbers come from public-domain manuals, which do not always agree: Dana (1847), Lever (1843) and The Bluejackets' Manual (1917) use reef knot and square knot for the same knot, while Verrill (1917) uses reef knot for a looped variant, and pages say which manual uses which name. A manual's claims keep their date: a 1917 manual's statement that a knot "does not slip" is reported as that manual's words, never as a safety rating.

The Ashley Book of Knots (1944) is cited by title only. Its US copyright was renewed in 1971 and runs to the end of 2039, and the site has no licensed access to the text, so each knot's reference line gives figure numbers from public-domain manuals instead.

The sources list on each page marks whether the full text or only an abstract was read, and a work known only from its catalog entry never supports a headline or a summary. Works in copyright are paraphrased, with at most a short quotation and its page number; public-domain manuals may be quoted. Strength and slip figures come only from named studies, with their test conditions. Where a holding institution's record could not be consulted, the page says so, cites what could be read and links the file page that was read.

Photographs, plates and illustrations

Every collection photograph and old-book plate is checked on its own, from the license on its file page, not the reputation of the institution that holds it. Public domain is written out per jurisdiction with its basis, for example "US: first published 1890; Germany and France: author died 1930", and an image is used only when it is public domain in both the United States and the European Union. Licenses that bar commercial use or changes are not used, and CC BY-SA images appear only in the body of a page, never as a card or share image. Images labeled Illustration are generated; they never stand in for a real object or show the structure of a knot, crease or stitch.

Safety, and what is never claimed

Every knot page gives a knot's limits and how it fails in the same paragraph that introduces it. No knot here is presented as able to hold a person, and nothing here teaches knots for climbing, rescue, work at height, lifting or towing; formal training and the standards for that work come first. No claim rests on how a knot or a fold feels in the hand, and a model is always called a model.

Data files

Tables and many diagrams come with a CSV file beside the article. Each row stands as one claim with its own source, and limits such as "approx" or "per the 1917 manual" sit in a qualifier column on the same row. The files hold facts and computed values, not text copied from works in copyright. When a page re-adds a historical table, the sums are computed from the printed readings, and groups that do not add up are listed as they are, with the original readings unchanged. Sizes measured on a scan are given as a share of the scanned page when the real size of that copy is not recorded. Reuse terms are on Attribution strings, and dated fixes are on Repairs.