INTERNATIONAL JOURNAL OF GEOMETRYVol. 9 (2020), No. 2, 69 - 80

FOLDING AXIOMS FOR A 3D EXTENSION

OF ORIGAMI

Jorge C. Lucero

Abstract. The axioms of origami are elementary fold operations per-formed on a sheet of paper which model the geometry of origami construc-tions. This article presents an extension of the axioms to the case of foldinga three-dimensional medium within a four-dimensional space. The axiomsare defined in terms of incidence constraints between given points, lines andplanes that must be satisfied by folding along a plane. A complete set of 36axioms is obtained and illustrated with examples.

1. Introduction

A convenient tool to study the geometry of origami constructions (i.e.,constructions by folding a sheet of paper or other material) is provided byits so-called axioms. The axioms are elementary fold operation along astraight line that satisfy specific alignments or superpositions between givenpoints and lines on a sheet of paper, and a total of seven operations havebeen found [8, 14]. The adequacy of the term “axioms” may be questionedsince some of the operations may be derived from others, and further, someof them may not be possible depending on the configuration of given pointsand lines. However, usage of the term is already well established in stud-ies on origami geometry and therefore is continued here. Studies using theaxioms usually regard the folded medium as an Euclidean plane or a por-tion of it, and folding on the sphere [16] and the hyperbolic plane [5] havebeen also considered. Further, extensions of the axioms to include multiplesimultaneous folds [8] and incidences with conics [6] have been studied.

Geometric constructions by folding are not restricted to the bi-dimensional(2D) case. They may be applied also to one-dimensional objects, as in thecase of linkages or chains [9], as well as extended to three dimensions (3D)by considering a 3D medium folded within a 4D or higher hyperspace. Forexample, 3D origami has been been applied to studies of Universe structurein cosmology for modeling the large-scale distribution of matter [20]. Otherstudies of 3D origami have considered its topology [21], foldability conditions[15], and visualization through computer graphics [13].

————————————–Keywords and phrases: Geometric Construction, Folding, Origami,

3D Space(2010)Mathematics Subject Classification: 51M15Received: 12.03.2020. In revised form: 24.06.2020. Accepted: 02.06.2020.

70 Jorge C. Lucero

P

Q

∆

Figure 1. Point Q is the reflection of point P in plane ∆ [18].

In recent papers, extensions of the folding axioms to the 3D case have beenproposed. In [12], a reformulation of seven 2D axioms to the case of foldingalong a plane (instead of a line) in a 3D space was presented. In [18], theaxioms were defined in terms of incidence constraints between points, lines,and planes that must be satisfied by folding along a plane. A total of 47 ax-ioms was found, although it was noted that some redundancy among themcould exist. In fact, the analysis included axioms for particular (non-generic)configurations of points, lines and planes that could be included within moregeneral axioms. In [7], a similar extension was presented; however, configu-rations involving non-parallel lines were disregarded. The analysis produced27 axioms, which constitute a subset of the larger set found in [18].

The present study has the purpose of reworking the analysis in [18] inorder to obtain a complete set of general and non-redundant 3D axioms.First, all possible incidence constraints for the fold plane will be reviewed,and next, a full list of axioms will be derived and expressed as folding oper-ations.

2. Definitions

Folds are modeled using the geometry of reflections [19]. In 2D origami,folding a sheet of paper along a straight line superposes the paper on oneside of the fold line to the other side. As a result, all objects (points andlines) on each side of fold line are reflected across it onto the other side. Asimilar model applies to the 3D extension, except that reflection takes placeacross a fold plane.

Given a plane ∆, the reflection F∆ in ∆ is the mapping on the set ofpoints in the 3D space such that for point P

F∆(P ) =

P if P ∈ ∆,Q if P /∈ ∆ and ∆ is the perpendicular bisector plane of

segment PQ,

(Fig. 1). Note that this is a symmetric mapping, i.e., F∆(P ) = Q iffF∆(Q) = P .

The reflection of a line m or plane π across the fold plane ∆, denoted asF∆(m) and F∆(π), respectively, is obtained by reflecting every point in mor π.

Folding axioms for a 3D extension of origami 71

In standard 2D origami constructions, the only allowed tool is the straightfold (curved fold origami is also possible, but this is an advanced techniqueand is not considered here; see, e.g., [11]). Therefore, the basic geometricobject is the straight line, which is assumed to be constructible anywhere inthe plane [10]. Points are next defined as the intersection of two non-parallellines. Then, the folding axioms are expressed in terms of superpositions orincidence constraints between points and lines that must be satisfied with afold [8, 14, 17]. Analogously, the basic geometric object in 3D origami is theplane, and lines are then defined as the intersection of non-parallel planes.Therefore, the definitions of axioms are extended to include incidences in-volving points, lines and planes [7, 18].

Each incidence involves an object α and the image F∆(β) of an object β(where α and β are not necessarily distinct) by reflection in the fold plane ∆.Considering three types of objects (point, line, and plane), and noting thatall incidence relations are symmetric (due to the symmetry of the reflectionmapping), then there are six possible object type attributions for α andβ, namely: point/point, line/line, plane/plane, point/line, point/plane andplane/plane. The six cases are considered in the next sections and, forconvenience, the cases in which an object is reflected onto itself are treatedseparately. A detailed analysis of each case is available in [18]; for clarity,main results and figures with examples are repeated here.

All given objects are assumed in general positions; e.g., : points are not onlines or planes, lines are not coplanar, lines are not parallel or perpendicular(normal) to planes, and planes are not parallel or normal between them.

3. Incidence constraints

3.1. Point/point. In this incidence, the reflection of a given point P coin-cides with another given point Q; i.e., F∆(P ) = Q, and we assume P 6= Q.Its solution is the perpendicular bisector of segment PQ (Fig. 1).

3.2. Line/line. In this incidence, the reflection of a given line m intersectsanother given line n, i.e. F∆(m) ∩ n 6= ∅, and we assume that m and n arenot coplanar. Its solution is a fold plane ∆ that is tangent to an hyperbolicparaboloid defined by the positions of m and n (Fig. 2). Two parametersare required to specify the point of tangency, and therefore the family ofsolutions has two degrees of freedom.

3.3. Plane/plane. In this incidence, the reflection of a given plane π co-incides with another given plane τ , i.e., F∆(π) = τ , and we assume that πand τ are not parallel. Its solution is a fold plane ∆ that bisects the dihe-dral angles between π and τ . There are two possible solutions and they areperpendicular to each other (Fig. 3).

3.4. Point/line. In this incidence, the reflection of a given point P is ona given line m, i.e., F∆(P ) ∈ m, and we assume P /∈ m. Its solution is afold plane ∆ that is tangent to a parabolic cylinder generated by a parabolain the plane containing the focus P and directrix m, translated in directionparallel to that plane (Fig. 4). Tangency between the fold plane and thecylinder occurs along a line normal to the plane of P and m. One parameter

72 Jorge C. Lucero

m

n

P

P ′ ∆

Υ

m′

Figure 2. Plane ∆ reflects m onto m′, which intersects nat P ′, and is tangent to the hyperbolic paraboloid Υ [18].

π

τ

∆1

∆2

Figure 3. The fold planes ∆1 and ∆2 reflects π onto τ [18].

is required to specify the tangency line, and therefore the family of solutionshas one degree of freedom.

3.5. Point/plane. In this incidence, the reflection of a given point P is ona given plane π, i.e, F∆(P ) ∈ π, and we assume P /∈ π. Its solution is afold plane ∆ that is tangent to a paraboloid generated by the rotation ofa parabola around its axis normal to π, and the parabola has focus P anddirectrix contained in π (Fig. 5). Two parameters are required to specifythe point of tangency, and therefore the family of solutions has two degreesof freedom.

3.6. Line/plane. In this incidence, the reflection of a given line m is on agiven plane π, i.e., F∆(m) ⊂ π, and we assume that m and π are not parallel.Its solution is a fold plane ∆ that is tangent to an elliptical cone defined bym and π, with vertex at their intersection (Fig. 6). Tangency between thefold plane and the cone occurs along a line passing through the vertex. Oneparameter is required to specify the tangency line, and therefore the familyof solutions has one degree of freedom.

Folding axioms for a 3D extension of origami 73

m

P

P ′

∆

Ξ

Figure 4. Plane ∆ reflects P onto P ′ ∈ m and is tangentto the parabolic cylinder Ξ [18].

π

P

P ′

∆

Ψ

Figure 5. Plane ∆ reflects P onto P ′ ∈ π and is tangent tothe paraboloid Ψ [18].

m

m′

π

∆

Ω

Figure 6. Plane ∆ reflects m onto m′ ∈ π and is tangent tothe elliptical cone Ω [18]. For clarity of the figure, only theupper nappe of Ω is shown.

74 Jorge C. Lucero

3.7. Point to itself. In this incidence, the reflection of a given point Pcoincides with itself, i.e., F∆(P ) = P . Its solution is a fold plane ∆ passingthrough P . An arbitrary normal direction for ∆ may be defined, e.g., by twoangles (azimuthal and polar) in a suitable coordinate system. Therefore, thefamily of solutions has two degrees of freedom.

3.8. Line to itself. In this incidence, the reflection of a line m coincideswith itself, i.e., F∆(m) = m. It has two families of solutions:

(1) A fold plane ∆ containing m reflects every point of m to itself. Aparticular fold plane may be specified by the dihedral angle froma reference plane; therefore, this family of planes has one degree offreedom.

(2) A fold plane ∆ perpendicular to m reflects every half (ray) of mto its opposite half. A particular fold plane may be specified byits distance from a reference point in m; therefore, this family ofsolutions also has one degree of freedom.

3.9. Plane to itself. In this incidence, the reflection of a plane π coincideswith itself, i.e., F∆(π) = π. Its solution is a fold plane ∆ perpendicular to π,which reflects half of π to its opposite half. A particular fold plane may bespecified by its angle from a reference direction, and distance to a referencepoint. Therefore, the family of solutions has two degrees of freedom

The incidence is also produced by reflecting every point of π to itself witha fold plane coincident with π. However, this folding does not creates a newplane and is therefore ignored. This is in consonance with the usual modelof 2D origami, where folding along a given line is not considered as a validaxiom because it does not create a new line [8, 14].

4. Axioms

A plane in 3D space is an object with three degrees of freedom. Thenumber of degrees of freedom that a particular fold plane ∆ consumes inorder to satisfy an incidence constraint is called the codimension of theconstraint [22]. Incidences in Sections 3.1 and 3.3 have either a unique or afinite number of solutions; therefore, they define constraints of codimension3. Incidences in Sections 3.4, 3.6 and 3.8 have solutions with two degreesof freedom and therefore they define constraints of codimension 2. Finally,incidences in Sections 3.2, 3.4, 3.7 and 3.9 have solutions with one degreeof freedom and therefore they define constraints of codimension 1. Table 1lists all the incidence constraints with their respective codimensions.

A folding axiom may be defined as a generically well-constrained foldplane. Let us recall that a well-constrained system is one that has a finitenumber of solutions, and the generic property means that the constraint pa-rameters are algebraically independent [22]. In the present case, the genericproperty is satisfied when all points, lines and planes involved in an axiomare given in general positions. A consequence of that property is that thefree parameters (if any) associated to all incidence constraints in an axiomare independent of each other. Therefore, the total codimension of the set isthe sum of the individual codimensions. To define an axiom, that sum mustbe equal to 3. If it is smaller than 3, then the system is under-constrained

Folding axioms for a 3D extension of origami 75

Table 1. Incidence constraints. P and Q are distinct points;m and n are distinct lines; π and τ are distinct planes.

Number Section Definition Codimension

1 3.1 F∆(P ) = Q 3

2 3.2 F∆(m) ∩ n 6= ∅ 1

3 3.3 F∆(π) = τ 3

4 3.4 F∆(P ) ∈ m 2

5 3.5 F∆(P ) ∈ π 1

6 3.6 F∆(m) ⊂ π 2

7 3.7 F∆(P ) = P 1

8 3.8 F∆(m) = m, where half of m is re-flected to its opposite half

2

9 3.8 F∆(m) = m, where every point of mis reflected to itself

2

10 3.9 F∆(π) = π, where half of π reflectedto its opposite half

1

and has infinite solutions. If it is larger than 3, then the system is over-constrained and has no solutions, or it may have non-generic solutions.

Each of incidences 1 and 3 already define an axiom, and the other inci-dences must be applied in combinations; i.e., an incidence of codimension 1and another of codimension 2, with a total of 16 possible combinations, orthree incidences of codimension 1, with a total of 20 possible combinations.

However, incidence 8 may not be applied together with incidence 10, sincethe combination would demand a fold plane normal to given line m and planeπ. That solution is possible only if m and π are parallel (i.e., non-generic).Even in that case, the associated constraints are mutually redundant andthe combination has an infinite number of solutions. Similarly, incidence10 may not be applied three times. That combination would demand a foldplane normal to three given planes, which is possible only if two of the planesare parallel.

The total number of axioms is therefore 36, and Table 2 lists them in theform of folding operations. Figs. 7 and 8 show examples of axioms 7 and32, respectively, from [18].

Table 2: 3D folding axioms.

Number Incidences Operation

1 1 Given points P and Q, fold along a plane to place Ponto Q.

2 3 Given planes π and τ , fold along a plane to place πonto τ .

3 2, 4 Given a point P and lines m, n and o, fold along aplane to place P onto m, and so that m intersects o.

Continued on next page

76 Jorge C. Lucero

Table 2 – Continued from previous page

Number Incidences Operation

4 2, 6 Given lines m, n and o, and a plane π, fold along aplane to place m onto π, and so that n intersects o.

5 2, 8 Given lines m, n and o, fold along a plane to reflecthalf of m to its opposite half, and so that n intersectso.

6 2, 9 Given lines m, n and o, fold along a plane containingm so that n intersects o.

7 4, 5 Given points P and Q, a line m and a plane π, foldalong a plane to place P onto m, and Q onto π.

8 5, 6 Given a point P , a line m and planes π and τ , foldalong a plane to place P onto π, and m onto τ .

9 5, 8 Given a point P , a line m and a plane π, fold alonga plane to place P onto π, and reflect half of m to itsopposite half.

10 5, 9 Given a point P , a line m and a plane π, fold along aplane containing m to place P onto π.

11 4, 7 Given points P and Q and a line m, fold along a planepassing through P to place Q onto m.

12 6, 7 Given a point P , a line m and a plane π, fold along aplane passing through P to place m onto π.

13 7, 8 Given a point P and a line m, fold along a plane pass-ing through P to reflect half of m to its opposite half.

14 7, 9 Given a point P and a line m, fold along a plane pass-ing through P and containing m.

15 4, 10 Given a point P , a line m and a plane π, fold along aplane to place P onto m, and to reflect half of π to itsopposite half.

16 6, 10 Given a line m and planes π and τ , fold along a planeto place m onto π, and to reflect half of τ to its oppo-site half.

17 9, 10 Given a line m and a plane π, fold along a plane con-taining m to reflect half of π to its opposite half.

18 2, 2, 2 Given lines m, n, o, p, q and r, fold along a plane sothat m intersects n, o intersects p, and q intersects r.

19 2, 2, 5 Given a point P , lines m, n, o and p, and a plane π,fold along a plane to place P onto π, and so that mintersects n, and o intersects p.

20 2, 2, 7 Given a point P , and lines m, n, o and p, fold along aplane passing through P so that m intersects n, and ointersects p.

21 2, 2, 10 Given lines m, n, o and p, and a plane π, fold alonga plane to reflect half of π to its opposite half, and sothat m intersects n, and o intersects p.

Continued on next page

Folding axioms for a 3D extension of origami 77

Table 2 – Continued from previous page

Number Incidences Operation

22 2, 5, 5 Given points P and Q, lines m and n, and planes πand τ , fold along a plane to place P onto π, Q onto τ ,and so that m intersects n.

23 2, 5, 7 Given points P and Q, lines m and n, and a plane π,fold along a plane passing through P to place Q ontoπ, and so that m intersects n.

24 2, 5, 10 Given a point P , lines m and n, and a plane π, foldalong a plane to place P onto π, and so that m inter-sects n.

25 2, 7, 7 Given points P and Q, and lines m and n, fold alonga plane passing through P and Q, and so that m in-tersects n.

26 2, 7, 10 Given a point P , lines m and n, and a plane π, foldalong a plane passing through P to reflect half of π toits opposite half, and so that m intersects n.

27 2, 10, 10 Given lines m and n, and planes π and τ , fold along aplane to reflect half of π to its opposite half, to reflecthalf of τ to its opposite half, and so that m intersectsn.

28 5, 5, 5 Given points P , Q and R, and planes π, τ and ρ, foldalong a plane to place P onto π, Q onto τ , and R ontoρ.

29 5, 5, 7 Given points P , Q and R, and planes π and τ and ρ,fold along a plane passing through P to place Q ontoπ, and R onto τ .

30 5, 5, 10 Given points P and Q, and planes π, τ and ρ, foldalong a plane to place P onto π, Q onto τ , and toreflect half of ρ to its opposite half.

31 5, 7, 7 Given points P , Q and R, and a plane π, fold along aplane passing through P and Q to place R onto π.

32 5, 7, 10 Given points P and Q, and planes π and τ , fold alonga plane passing through P to place Q onto π, and toreflect half of τ to its opposite half.

33 5, 10, 10 Given a point P and planes π, τ and ρ, fold along aplane to place P onto π, and to reflect half ofτ to itsopposite half, and half of ρ to its opposite half.

34 7, 7, 7 Given points P , Q and R, fold along a plane passingthrough them.

35 7, 7, 10 Given points P and Q, and a plane π, fold along aplane passing through P and Q to reflect half of π toits opposite half.

36 7, 10, 10 Given a point P and planes π and τ , fold along a planepassing through to reflect half of π to its opposite half,and half of τ to its opposite half.

78 Jorge C. Lucero

m

P

Q

π

∆

Ξ

Ψ

Figure 7. Example of axiom 7 [18]. The fold plane ∆ istangent to the parabolic cylinder Ξ, with focus P and direc-trix line m, and the paraboloid Ψ, with focus Q and directrixplane π.

π

P

Q

∆

Ψ

τ

Figure 8. Example of axiom 32 [18]. The fold plane ∆contains point Q, is tangent to paraboloid Ψ with focus Pand directrix plane π, and is normal to plane τ .

5. Discussion

The set of axioms in Table 2 is complete, since all possible incidenceconstraints have been considered. It is also non-redundant, in the sensethat axioms involving the same number and types of objects have differentsolutions. An exhaustive analysis shows that there only three of such groupsof axioms: (a) axioms 5 and 6, (b) axioms 13 and 14, and (c) axioms 9, 10,12 and 15. The first group involves three given lines, and solutions to theaxioms are fold planes tangent to an hyperbolic paraboloid defined by twoof the lines (see Fig. 2), and also normal (axiom 5) or containing (axiom 6)the third line. The respective constructions for each axiom produce differentfold planes; even if the same lines are selected to define the same hyperbolic

Folding axioms for a 3D extension of origami 79

n

m

P

∆

Figure 9. The fold plane ∆ contains m and is perpendicularto n.

paraboloid in both axioms, the fold planes for each axiom still will be normalbetween them. A similar analysis applies to the other two groups.

The 36 axioms constitute a subset of those found in [18]. The later setalso contained axioms for non-generic configurations of objects, as thoseobtained by combining two incidences both of codimension 2 and whichdo not fit the definition adopted here. Further, the solutions to those non-generic configurations may be also obtained by applying generic axioms. Forexample, the combination of incidences 8 and 9 demands a fold plane thatcontains a given line m and is perpendicular to another given line n, and asolution exists only if m and n have perpendicular directions (see Fig. 9).However, the same fold plane is obtained by selecting any point P ∈ m andapplying axiom 13 on P and n.

Next steps in the study of 3D origami axioms should determine the num-ber of solutions of each of them, and conditions for the existence of thosesolutions, including non-generic cases. Also, the present analysis may be ex-tended to higher dimensions, where an (n−1)-dimensional “sheet” is foldedwithin an n-dimensional space. In that case, folding takes place along afold hyperplane, and axioms are defined in terms of incidence constraintsbetween affine subpaces.

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DEPARTMENT OF COMPUTER SCIENCEUNIVERSITY OF BRASILIABRASILIA, DF, 70910-900, BRAZILE-mail address: lucero@unb.br