Constructions

A construction is the read of sort line that a write's axis comes from: a set of alignments that together determine one fold line, or several. An alignment is an incidence between two objects, each a point, a line, or the folded image of one: a point onto a point, a point onto a line, a line onto a line, the fold line through a point, the fold line perpendicular to a line. Alperin and Lang show that the seven Huzita-Justin axioms are exactly the minimal sets of such alignments that determine one fold line with finitely many solutions ¹, and that the same alignments, distributed over two fold lines, give the 489 two-fold axioms ². The construction with its alignments is the canonical form; an axiom number is the name of one such set.

(align (.a onto .c))                            ; axiom 2
(align (through .a) (through .b))               ; axiom 1
(align (perp --l) (through .p))                 ; axiom 3
(align (.p onto --l) (through .q)) (toward .a)  ; axiom 6, and the side that stays
(align (.p onto --l) (perp --m))                ; axiom 4
(align (--l onto --m)) (toward .c)              ; axiom 5, and the side that stays
(align (.a onto --m) (.c onto --l))             ; axiom 7

2 assertions passed

assert .p = (1/2, 1/2)

assert .q = (1/2, 0)

The prose forms of the seven axioms are sugar for these, the prose_axiom alternatives: (through .a .b), (map .a onto .c), (perp --l through .p), (map .p onto --l perp --m), (map --l onto --m), (map .p onto --l through .q), (map .p onto --l and .q onto --m). A prose form means what its align spelling means. The axiom number names one alignment set, recognized by the kinds of its alignments; it appears in error messages, in provenance ("axiom": "axiom6" in beloch:edges) and here, and it is no type of its own. A set that is none of the seven is an error naming its alignments.

A construction holds what determines the line: its alignments and its heading. What determines the fold, toward, moving and the placement, are items of the statement that reads the construction (ADR 0031). A binding takes the side items as a write does, --k = (map --v onto --h) (toward .b), so a bound line is selected the way a fold along it would be. A construction inside an expression takes no selection; it is bound first. A toward written inside a construction is an error, with the hint write it as an item of the write: (toward .p).

A construction over several fold lines names them and lets each alignment say which line folds: (align --a --b (--a .p onto --l) (--b .q onto --m) (--a .r onto --b .s)) is the two-fold axiom Alperin and Lang call AL6ab8. Its name is composed from the kinds of its alignments, as every two-fold name is ³; packages/multifold computes the same names. No two-fold construction is evaluated by the kernel yet; the syntax is fixed here so that it needs no second form when one is.

Open 1 naming fold lines inside a construction

The two-fold form above writes the folding line in front of the object it folds, (--a .p onto --l), and a doubly folded alignment as (--a .r onto --b .s). Whether the fold lines of a construction are declared in the align head, as above, or are the names the binding form gives, and how an alignment on a virtual point (Alperin and Lang's AL10) is written, is not decided and waits for the first two-fold construction the kernel can solve.

Axiom numbering

The axioms carry the numbers of Justin's table ⁴, which orders the seven operations by algebraic power: the number of an axiom is the rung of the number ladder its construction needs (ADR 0012). Two other schemes are in use and line up with neither Justin's nor each other; the table maps them.

Beloch, Justin	the fold	degree	solutions	Huzita-Hatori	Hull ⁵
1	the line through two points	1	1	1	O1
2	one point onto another	1	1	2	O3
3	the perpendicular through a point	1	1 or 2	4	O5
4	a point onto a line, perpendicular to another line	1	0 or 1	7	O8
5	one line onto another	2	1 or 2	3	O4
6	a point onto a line, through a point	2	0 to 2	5	O6
7	two points each onto a line, the Beloch fold	3	0 to 3	6	O7
	the meet of two lines, no fold				O2

Huzita-Hatori, the numbering Wikipedia uses, numbers the cubic fold 6. Beloch numbers it 7, and "axiom 7" on these pages is the Beloch fold.
Hull inserts the meet of two lines as his O2 and orders 3, 4 and 5 differently, so point onto point is axiom 2 here and Hull's O3, and the angle bisector is axiom 5 here and Hull's O4.
Axioms 1 to 4 have degree 1 and stay in the rationals; axiom 5 is the first that needs a square root, and axiom 7 the first that needs a cube root. In Justin's account axioms 1 to 4 give the dyadic rationals, 1 to 5 the field 1 to 6 the ruler-and-compass field and 1 to 7 the larger 0@justin1986, §8.1].

The clashes with the other two schemes are also recorded in Dead ends & antipatterns.

Selection

Axioms 5, 6 and 7 leave finitely many candidate lines, usually more than one, and the statement that reads the construction keeps one of them (def-selection). A candidate creases a face of the current state; a line off the paper, or one grazing an edge or a corner, creases nothing and is no candidate. An onto alignment is an incidence: after the fold, one of its two objects lies on the other, and either may be the one that moves, so (--l onto .p) means (.p onto --l). The stages run in order, numbered as in the model, and each error below names the stage that left no line or several.

(heading --l) keeps the candidates whose line makes the smallest angle with --l. The position of --l plays no part, and a tie passes on to toward.
(toward …) names the side of each candidate that stays: the side that holds the point, or the side the line lies on. A point on the candidate, or a line the candidate crosses on the paper, names no side of it. (moving …) names the side that folds over: the side of its first anchor point off the candidate, or the side of a line's material. With both items, moving has to name the side opposite toward.
A candidate remains when the fold with that side carries out every alignment: one of its objects lies on the side that folds over and lands on the paper of the other, or a point lies on the candidate and on its target line already. A point that folds onto a line lands on the part of the line that lies on the paper, and a line that folds onto a point has paper at the place that lands on it. A landing on a line's extension beyond the edge cannot be made by eye; a program that wants one constructs the target line first. A candidate for which stage 2 names no side does not remain. With neither item, a candidate remains when either side carries the construction out.
Among several remaining candidates, toward keeps the one that lands the objects of its alignments nearest it: the points and the parts of lines that fold over, where they land, or with (x toward …) only the part of x. The rest of the flap goes unmeasured. Distances are between sets and compared exactly.

The side that stage 2 names is the side the fold moves. With no side item, the side that alone carries the construction out folds over, and where both do, the side of the first object as the program writes it: fold (map .a onto .b) folds .a onto .b, and fold (map .a onto .b) (toward .a) folds .b onto .a. Where that object names no side either, a line lying across the candidate or a point on it, the write needs toward or moving. toward names the side that stays on every write: on a construction of one line, on a fold along an existing crease, fold (--v) (toward .b), on reverse (--h) (toward .d), whose tip lies opposite .d, and on a placed fold, whose block is the material opposite the toward where the write has no anchor.

fold (map .a onto .b) (toward .a)

1 assertion passed

assert .b = (0, 0)

(x toward .u) names what goes toward .u. x is one of the objects of the construction's alignments, compared by place, and the candidates that do not fold x over drop out at stage 4, where x is measured, also when no other candidate is left to compare.

Errors, with the stage that raises them:

no candidate crosses the paper: "map .p onto --d through .q: no crease lands on the paper, so there is no fold to make";
no candidate carries out the construction: with toward, "no fold of map .p onto --d through .q keeps .x on the side that stays and carries out its alignments"; with moving alone, "no fold of … moves .c and carries out its alignments"; with neither, "no fold of … carries out all its alignments at once";
toward and moving naming the same side: "toward .x and moving .c name the same side of the fold line", hint "drop one of them";
a toward on a line that determines one: "toward .x names no side of the fold line";
(x toward …) with an x that is no object of the construction: "x is none of the objects of …";
two candidates landing equally near toward: "toward .x lies as near to where one fold of … lands the objects of its alignments as to where another does", with (x toward …) naming x in place of the objects, hint "name what goes toward .x, e.g. (.p toward .x)";
several candidates remaining without toward: "map .p onto --d through .q is ambiguous: 2 folds carry it out, all landing on the paper", hint "add (toward .x) or a heading";
heading on a construction that determines one line, a parallel axiom 5 among them: "this construction determines one line", hint "drop heading";
toward or moving on a mark along an existing line, which selects nothing and moves nothing.

When several candidates remain, the trace (FOLD.md, "The trace") carries one suggestion per remaining candidate: an item that, added to the statement, keeps that candidate alone. Without toward, the selection runs again with (toward x) for each named point in scope and then each named line, in program order, and the suggestion is the first x after which exactly that candidate remains. After a tie at stage 4 of a bare toward, it tries (x toward …) over the objects of the alignments in the order the program writes them. A candidate no such item keeps alone has no suggestion, and a selection that ends with no candidate has none. A suggestion is checked for its own statement only; a later statement that fails with it reports that failure where it happens.

Axiom 1, the line through two points

(through .p .q) is the line through .p and .q ⁶, ⁷: one line, and nothing to select. Its one error is two points at one place, the same name written twice or two material points a fold has brought together (Two points at one place).

mark (through .a .c) as --ac
mark (through .b .d) as --bd
.o = --ac * --bd

1 assertion passed

assert .o = (1/2, 1/2)

Axiom 2, a point onto a point

(map .p onto .q) folds .p onto .q: the perpendicular bisector of the two ⁸, ⁹. One line, and the fold it carries derives its moving flap from .p (Write statements). Its error is the same as for axiom 1.

mark (map .a onto .c) as --f
mark (through .a .c) as --ac
.o = --f * --ac

1 assertion passed

assert .o = (1/2, 1/2)

Axiom 3, the perpendicular through a point

(perp --l through .p) is the line through .p perpendicular to --l ¹⁰, ¹¹, the first axiom that takes a line as an operand. Justin's operation ③ admits two folds when .p lies on --l, the perpendicular and --l itself, which scores the line again and makes nothing; Beloch takes the perpendicular in every case, so the construction has no error of its own.

mark (through .a .c) as --ac
mark (perp --ac through .b) as --p
.o = --p * --ac

1 assertion passed

assert .o = (1/2, 1/2)

mark (through .a .c) as --ac
.o = free on --ac from .a at 1/2
mark (perp --ac through .o) as --p    ; .o lies on --ac
.x = --p * --ab

1 assertion passed

assert .x = (1, 0)

Axiom 4, a point onto a line along another

(map .p onto --l perp --m) folds .p onto --l with a crease perpendicular to --m: .p moves parallel to --m until it lands on --l ¹², ¹³. At most one line, so there is nothing to select, and toward still names the side that stays (Selection, above). When .p lies on --l already, the crease is the perpendicular to --m through .p. The result stays rational ¹⁴. Its error is --l parallel to --m, map .p onto --l perp --m: lines are parallel, no fold exists, which covers the case of no landing and the case of .p on --l with every perpendicular a fold.

mark (map .a onto --bc perp --ab) as --v   ; .a moves along --ab onto --bc
.x = --v * --ab

1 assertion passed

assert .x = (1/2, 0)

mark (through .a .c) as --ac
.m = free on --ab from .a at 1/2
mark (map .m onto --ab perp --ac) as --p  ; .m lies on --ab already
.x = --p * --da

1 assertion passed

assert .x = (0, 1/2)

Axiom 5, a line onto a line

(map --l1 onto --l2) folds --l1 onto --l2 along an angle bisector ¹⁵, ¹⁶. Two lines that cross have two bisectors, perpendicular to each other. Two parallel lines have one midline, and two identical lines are an error, "lines are identical". A line's material is the segments of a crease, or the chord of the paper under a line bound by =. The kite base folds an edge onto the long diagonal; one bisector crosses the paper and the other touches it only at .a, so only the first is a candidate.

mark (through .a .c) as --ac
fold (map --da onto --ac)

1 assertion passed

assert steps = 1

Where the two lines cross inside the material of --l1, each bisector has material of both lines on both sides. (--l1 toward .x) measures the part of --l1 on the side away from .x: the two folds land it on the two rays of --l2 from the crossing, and the nearer landing is kept. Two landings on the same ray, as long as each other, are equally near every point, and so are two landings on opposite rays for a point on the perpendicular to --l2 through the crossing (lem-crossing-landing). A bare toward measures the part of --l2 that folds over too, and ties more often. moving then separates the candidates by the flap it names. A fold with no side item whose --l1 lies across the fold line is an error, "--l1 straddles the fold line", hint "add moving to pick the swinging flap".

mark (through .a .c) as --ac
mark (through .b .d) as --bd
fold (map --ac onto --bd) (toward .b) (moving .c)

1 assertion passed

assert .c = .b

The bisector of two rational lines is irrational in general, slope $\sqrt2 - 1$ for $y = 0$ and $y = x$ ; equality, parallelism and the paper tests stay exact (Exactness).

Axiom 6, a point onto a line through a point

(map .p onto --d through .q) folds .p onto --d with a crease through .q ¹⁷. .q lies on the crease, so it is as far from .p as from the landing on --d, which lies on the circle about .q through .p: up to two landings, and a crease for each, the perpendicular bisector of .p and its landing. Square roots suffice; cube roots first appear with axiom 7.

The bird base's kite crease is the case the paper settles: on the preliminary base, map .sr onto --mid through .c has two candidates, and the outward one meets the folded base only at .c (packages/core/tests/cases/bases/bird-base.bel). The largest equilateral triangle in the square needs toward, since both candidates cross the paper (fig-candidates).

Errors of its own: .p and .q the same point, "…: .p and .q are the same point, so no fold exists"; the circle missing --d, "cannot fold .p onto --d through .q: out of reach". When .p lies on --d already, the identity landing is dropped and the mirror landing gives the crease.

Axiom 7, two points onto two lines

(map .p onto --d and .q onto --e) folds .p onto --d and .q onto --e with one crease ¹⁸. Each alignment traces a parabola with focus the point and directrix the line, and the crease is a common tangent of the two, a root of a cubic: up to three candidates ¹⁹. This is the operation that doubles the cube and trisects an angle, beyond ruler and compass. The crease is irrational in general, its coordinates in a field $\mathbb{Q}(\alpha)$ with $\alpha$ algebraic of degree 3, and they are compared exactly.

A candidate with .p and .q on opposite sides still carries out both alignments when the side of .p folds over and has paper of --e at the place that lands on .q: .p lands on --d, and --e lands on .q. (.p toward .x) measures where .p lands.

mark (through .a .c) as --diag
mark (through .b .d) as --anti
fold (map .a onto --anti and .d onto --diag) (.a toward .b)

2 assertions passed

assert .a = (1, 0)

assert .d = (1, 1)

Errors of its own: .q on --e already, "…: .q already lies on --e", with a hint to use axiom 6 and axiom 4 instead; --d parallel to --e, "…: --d and --e are parallel, so the cubic degenerates and no fold exists"; no common tangent, "cannot fold .p onto --d and .q onto --e: out of reach (no common tangent)". When .p lies on --d already, the cubic has a root where .p lands on itself; as in axiom 6, that identity landing is dropped.

Notes

Roger C. Alperin and Robert J. Lang, “One-, Two-, and Multi-Fold Origami Axioms,” in Origami⁴: Fourth International Meeting of Origami Science, Mathematics, and Education, ed. Robert J. Lang (A K Peters, 2009), sec. 3.↩
Roger C. Alperin and Robert J. Lang, “One-, Two-, and Multi-Fold Origami Axioms,” in Origami⁴: Fourth International Meeting of Origami Science, Mathematics, and Education, ed. Robert J. Lang (A K Peters, 2009), sec. 4.↩
Roger C. Alperin and Robert J. Lang, “One-, Two-, and Multi-Fold Origami Axioms,” in Origami⁴: Fourth International Meeting of Origami Science, Mathematics, and Education, ed. Robert J. Lang (A K Peters, 2009), sec. 4.↩
Jacques Justin, “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques,” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): sec. 8.1.↩
Thomas C. Hull, Origametry: Mathematical Methods in Paper Folding (Cambridge University Press, 2020), sec. 1, https://doi.org/10.1017/9781108778633.5.↩
Jacques Justin, “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques,” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): sec. 8.1,.↩
Thomas C. Hull, Origametry: Mathematical Methods in Paper Folding (Cambridge University Press, 2020), sec. 1, https://doi.org/10.1017/9781108778633.5.↩
Jacques Justin, “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques,” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): sec. 8.1, operation ②.↩
Thomas C. Hull, Origametry: Mathematical Methods in Paper Folding (Cambridge University Press, 2020), sec. 1, https://doi.org/10.1017/9781108778633.5.↩
Jacques Justin, “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques,” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): sec. 8.1, operation ③.↩
Thomas C. Hull, Origametry: Mathematical Methods in Paper Folding (Cambridge University Press, 2020), sec. 1, https://doi.org/10.1017/9781108778633.5.↩
Jacques Justin, “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques,” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): sec. 8.1, operation ④.↩
Thomas C. Hull, Origametry: Mathematical Methods in Paper Folding (Cambridge University Press, 2020), sec. 1, https://doi.org/10.1017/9781108778633.5.↩
Jacques Justin, “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques,” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): sec. 8.2(a).↩
Jacques Justin, “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques,” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): sec. 8.1, operation ⑤.↩
Thomas C. Hull, Origametry: Mathematical Methods in Paper Folding (Cambridge University Press, 2020), sec. 1, https://doi.org/10.1017/9781108778633.5.↩
Jacques Justin, “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques,” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): sec. 8.1, operation ⑥.↩
Jacques Justin, “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques,” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): sec. 8.1, operation ⑦.↩
Thomas C. Hull, Origametry: Mathematical Methods in Paper Folding (Cambridge University Press, 2020), sec. 2, https://doi.org/10.1017/9781108778633.3–2.4.↩

References

Alperin, Roger C., and Robert J. Lang. “One-, Two-, and Multi-Fold Origami Axioms.” In Origami⁴: Fourth International Meeting of Origami Science, Mathematics, and Education, edited by Robert J. Lang. A K Peters, 2009. ↑ a b c

Hull, Thomas C. Origametry: Mathematical Methods in Paper Folding. Cambridge University Press, 2020. https://doi.org/10.1017/9781108778633. ↑ a b c d e f g

Justin, Jacques. “Résolution Par Le Pliage de l’équation Du Troisième Degré et Applications Géométriques.” L’Ouvert (Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg), no. 42 (March 1986): 9–19. ↑ a b c d e f g h i