aa r X i v : . [ m a t h . L O ] D ec Axiomatizing Origami planes
L. Beklemishev , A. Dmitrieva , and J.A. Makowsky Steklov Mathematical Institute of RAS, Moscow, Russia National Research University Higher School of Economics, Moscow University of Amsterdam, Amsterdam, The Netherlands Technion – Israel Institute of Technology, Haifa, IsraelDecember 8, 2020
Abstract
We provide a variant of an axiomatization of elementary geometrybased on logical axioms in the spirit of Huzita–Justin axioms for theOrigami constructions. We isolate the fragments corresponding to nat-ural classes of Origami constructions such as Pythagorean, Euclidean,and full Origami constructions. The sets of Origami constructiblepoints for each of the classes of constructions provides the minimalmodel of the corresponding set of logical axioms.Our axiomatizations are based on Wu’s axioms for orthogonal ge-ometry and some modifications of Huzita–Justin axioms. We work outbi-interpretations between these logical theories and theories of fieldsas described in J.A. Makowsky (2018). Using a theorem of M. Ziegler(1982) which implies that the first order theory of Vieta fields is unde-cidable, we conclude that the first order theory of our axiomatizationof Origami is also undecidable.
Dedicated to Professor Dick de Jongh on theoccasion of his 81st birthday
Planar Origami geometry has been studied from an axiomatic mathematicalpoint of view since the 1980s, when J. Justin first proposed an axiomatization1n [Jus89]. In [Mak18, Mak19] a proof was outlined of the statement that thefirst order theory of Origami planes was undecidable. Although the proofstrategy is feasible, the exact definition of the first order theory of Origamiplanes was left imprecise. In particular, the role of the betweenness relationwas overlooked. The purpose of this paper is to provide a precise definitionof the first order theory of Origami planes and to establish its properties, assuggested in [Mak19].J. Justin identified six axioms (H-1), . . . , (H-6), later called the Huzita– Justin or Huzita – Hatori axioms, which capture the essence of Origamiconstructions [Wikb]. These axioms were not meant to be understood asaxioms in a logical sense but rather as specifying a (not necessarily deter-ministic or always defined) set of operations generating the ‘Origami con-structible’ points on the plane. Along these lines, Alperin [Alp00] charac-terized the classes of points (corresponding to certain subfields of C ) con-structible using Origami constructions defined by natural subsets of Huzita–Justin operations. Our paper uses the ideas of Alperin in the part that dealswith the interpretations between geometric theories and the theories of thecorresponding classes of fields.As it turns out, if one directly translates Huzita – Justin axioms intoa logical language treating the free variables in these axioms as universallyquantified, the axioms (H-5) and (H-6) become inconsistent. Therefore, thequestion arises, how we can adapt the principles of Origami constructionsto serve as logical axioms of planar geometry. There are two issues here:what are the basic predicates, in which the the axioms (H-1), . . . , (H-6) canbe formulated, and what are the additional axioms which govern the basicpredicates.Looking at the Huzita – Justin axioms it appears as if one could axiom-atize Origami geometry in the language of incidence and orthogonality only,taking as the underlying geometry a metric Wu plane. However, it turnsout that there are multiple reasons for choosing as the underlying geometryan ordered metric Wu plane.Firstly, the Huzita – Justin axiom (H-3), which says that for any twolines l , l there is a fold which places l on l , is only true if l , l are non-isotropic (not orthogonal to themselves). Our formulation of (H*3) takesthis into account.Secondly, Axiom (H-5) states that, given two points P and P and a line ℓ , one should be able to construct a fold that places P onto ℓ and passesthrough P . However, such a fold only exists provided P is closer to the The seventh axiom was later shown to be superfluous. ℓ than P . This condition needs to be expressible in the language,which we achieve by introducing the betweenness relation, as formulated in(H*5). Other possibilities are discussed at the end of the paper.We say that a metric Wu plane is orderable if it can be equipped witha ternary relation Be ( A, B, C ) which satisfies the Hilbertian axioms of be-tweenness. In Proposition 3 we prove that a metric Wu plane is orderableiff it has no isotropic lines.If
A, B, C are colinear on a line ℓ , let ℓ ′ be orthogonal to ℓ going throughthe point C and let B ′ be the point on ℓ obtained by placing B on ℓ after fold-ing along ℓ ′ . In the real plane and the presence of the betweenness relation B is between A and C , Be ( A, B, C ) we have
Out ( A, C, B ) iff either Be ( A, B, C )or Be ( A, B ′ , C ). Therefore Out ( A, C, B ) is definable using Be ( A, B, C ) andthe usual axioms for betweenness. It is not obvious, however, if Be ( A, B, C )can be formulated using an axiomatization of
Out ( A, C, B ). The same canbe said about
Closer ( A , A , ℓ ).A reasonable axiomatization of Origami geometry can be obtained fromordered metric Wu planes by adding a finite set of axioms.Metric Wu planes already satisfy Huzita – Justin axioms (H-1), (H-2),(H*3), (H-4) where (H*3) is our modification of (H-3), see Proposition 1.An ordered Origami plane is an ordered metric Wu plane which satisfies alsoour modified axioms (H*5) and (H*6). This deviates from the definitiongiven in [Mak18, Mak19] where an ordered Origami plane is defined in aninconsistent way.The ordered metric Wu planes are bi-interpretable with ordered Pythagoreanfields, whose first-order theory is undecidable by Ziegler’s theorem. Simi-larly, we obtain that the first order theory of our axiomatization of Origamigeometry is also undecidable.In the next section we give background and outline our main results.The axioms mentioned in this section are given explicitly in Section 3 forWu planes, in Section 6 for the first four Huzita – Justin axioms, and inSection 7 for the betweenness axioms. In Section 4 we give the details ofthe coordinatization of Wu planes. This is an expanded version of the corre-sponding section of [Mak19]. In Section 5 we state a general undecidabilityresult for geometric theories of metric Wu planes. In Section 6 we showthat metric Wu planes satisfy the first four Huzita – Justin axioms. In Sec-tion 7 we introduce ordered metric Wu planes and their relation to orderedPythagorean fields. In Section 8 we introduce Euclidean and Vieta fields andprove our main Theorem 2. Finally, in Section 9 we discuss the remainingopen questions. 3 Background and main results
Our axiomatization uses two basic sorts of variables, denoting lines andpoints, respectively. We use
P, P , . . . , P i , and more liberally, upper caseletters, to denote points and ℓ, ℓ , . . . , ℓ i , and more liberally, lower case let-ters, to denote lines.In our axiomatization we use the following basic relations:(i) the incidence relation P ∈ ℓ between points and lines,(ii) the orthogonality relation ℓ ⊥ ℓ between two lines,(iii) and the betweenness (or order ) relation between three points P , P , P denoted by Be ( P , P , P ).The sorts are, however, definable using the incidence relation: ℓ is aline iff ∃ P ( P ∈ ℓ ), and P is a point iff ∃ ℓ ( P ∈ ℓ ). We also note that theequidistance relation P Q ∼ = RS between points P, Q and
R, S is definablefrom incidence and orthogonality [Wu94, page 25].We start with Wu’s axiomatization of orthogonal geometry, cf. [Wu94],augmented by the axioms for betweenness, and appropriate versions ofHuzita–Justin axioms. We denote by τ wu the vocabulary corresponding to(i)-(ii) and by τ o − wu the vocabulary corresponding to (i)-(iii). By FOL( τ wu )and FOL( τ o − wu ) we denote the corresponding sets of first order formulas.The Huzita – Justin axioms (H-1), (H-2), (H-4) can be used as they are.However, we modify axioms (H-3), (H-5), (H-6) to (H*3), (H*5), (H*6) ina way to assure that they are always applicable. The axioms (H-1), (H-2),(H*3), (H-4), (H*6) can be formulated in FOL( τ wu ). In order to formulate(H*5) we use the betweenness relation.As the most general class of structures for our study we consider metricWu planes (see [Mak19]). Metric Wu planes already satisfy Huzita – Justinaxioms (H-1), (H-2), (H*3), (H-4), where axiom (H-3) is modified so that itholds more generally in arbitrary metric Wu planes rather than just in theordered ones. Then we add to the axioms of metric Wu planes the axiomsof betweenness and further Origami axioms (H*5), (H*6) and obtain thecorresponding classes of structures. See Section 3 for the list of axioms usedin the definitions below. Definition 1. (i) A τ wu -structure Π is a metric Wu plane if it satisfies (I-1), (I-2), (I-3),(O-1), . . . , (O-5), the axiom of infinity (InfLines), (ParAx), the twoaxioms of Desargues (De-1) and (De-2) and (AxSymAx).4ii) An ordered metric Wu plane is a metric Wu plane satisfying axioms oforder (B-1), . . . , (B-4).(iii) A Euclidean ordered metric Wu plane is an ordered metric Wu planesatisfying (H*5).(iv) An ordered Origami plane is an ordered metric Wu plane satisfying(H*5) and (H*6).Only items (iii) and (iv) involve Origami or modified Origami axioms intheir definitions.The following theorems are from J.A. Makowsky [Mak19]. They can beseen as a logical formalization of the classical results on coordinatization(see [Hal43, Wu94]).
Theorem 1. (i) The theory of metric Wu planes is bi-interpretable withthe theory of Pythagorean fields of characteristic .(ii) The theory of ordered metric Wu planes is bi-interpretable with thetheory of ordered Pythagorean fields. Ziegler [Zie82] showed that any finitely axiomatized subtheory of thefirst order theory of real closed fields is undecidable. One of the main ideasof [Mak19] was the use of this result (via bi-interpretability) to show thatcertain elementary theories of geometries are undecidable. For example, asa corollary of Ziegler’s theorem one can conclude from Theorem 1 that theelementary theories of the classes of metric Wu planes and of ordered metricWu planes are undecidable.The main contributions of this paper are as follows.
Theorem 2. (i) The theory of Euclidean ordered metric Wu planes isbi-interpretable with the theory of Euclidean fields;(ii) The theory of ordered Origami planes is bi-interpretable with the theoryof Vieta fields.
Again, via bi-interpretability, Ziegler’s theorem is applicable to boththeories, hence both are undecidable.
Definition 2.
Pythagorean field is a field for which every sum of twosquares is a square: ∀ x, y ∃ z x + y = z τ ∈ consisting justof the incidence relation. Hilbert’s axioms of incidence(I-1):
For any two distinct points
A, B there is a unique line ℓ with A ∈ ℓ and B ∈ ℓ . (I-2): Every line contains at least two distinct points. (I-3):
There exist three distinct points
A, B, C such that no line ℓ containsall of them. Hilbert’s (sharper) axiom of parallels(ParAx):
For each point A and each line ℓ there is at most one line ℓ ′ with ℓ k ℓ ′ and A ∈ ℓ ′ . Axiom schema of infinity and Desargues’ axioms(InfLines):
Given distinct
A, B, C and ℓ with A ∈ ℓ , B, C ℓ we constructa line ℓ going through C and parallel to AB and define A as theintersection of ℓ and ℓ . Inductively, we define ℓ n as a line goingthrough C and parallel to A n B and define A n +1 as its intersectionwith ℓ . Then all the A i are distinct. (De-1): If the three pairs of the corresponding sides of two triangles
ABC and A ′ B ′ C ′ are all parallel to each other, i.e., AB k A ′ B ′ , AC k A ′ C ′ , BC k B ′ C ′ , then the three lines AA ′ , BB ′ , CC ′ joining thecorresponding vertices of these two triangles are either parallel to eachother or concurrent. (De-2): If two pairs of the corresponding sides of two triangles
ABC and A ′ B ′ C ′ are parallel to each other, say AB k A ′ B ′ , AC k A ′ C ′ , andthe three lines joining the corresponding vertices are distinct yet ei-ther concurrent or parallel to each other, then the third pair of thecorresponding sides are also parallel to each other, i.e., BC k B ′ C ′ .6 efinition 3. A τ ∈ structure Π is a Desarguesian plane if it satisfies (I-1,I-2, I-3), the axiom of infinity (InfLines), (ParAx) and the two axioms ofDesargues (De-1) and (De-2).Now we introduce a new relation of orthogonality ⊥ and consider thelanguage τ wu consisting of ∈ and ⊥ . Orthogonality axioms(O-1): ℓ ⊥ ℓ iff ℓ ⊥ ℓ . (O-2): Given O and ℓ , there exists exactly one line ℓ with ℓ ⊥ ℓ and O ∈ ℓ . (O-3): If ℓ ⊥ ℓ and ℓ ⊥ ℓ then ℓ k ℓ . (O-4): For every O there is an ℓ with O ∈ ℓ and ℓ ℓ . (O-5): The three heights of a triangle intersect in one point.Concerning Axiom (O-4) we remark that lines ℓ such that ℓ ⊥ ℓ arecalled isotropic . Caveat:
Without any axioms of order, the axioms of metric Wu planesdo not exclude the existence of isotropic lines.
Definition 4. A τ wu structure Π is an orthogonal Wu plane if it is a De-sarguesian plane satisfying orthogonality axioms (O-1, O-2, O-3, O-4, O-5). Axiom of symmetric axis.
We assume that we are working in an or-thogonal Wu plane. In order to formulate the next axiom, we follow [Wu94,page 22, Definition 3] to define the relation of being a symmetric point usingonly the Incidence relation.For two arbitrary points A = B on a line ℓ , take an arbitrary E / ∈ ℓ and construct a line ℓ ′ parallel to ℓ such that E ∈ ℓ ′ . Let D be theintersection of ℓ ′ and a line going through B parallel to AE . Then ABDE is a parallelogram. Finally, construct C as the intersection of ℓ and theline going through D and parallel to EB . Then, due to Desargues’ axioms,point C is independent of the construction of ABDE . We say that C is the symmetric point of A with respect to B . In addition, any point is said to bea symmetric point of the point with respect to itself. If C is the symmetricpoint of A with respect to B , we call B the midpoint of A and C . Then forany A and C their midpoint always exists and is unique.7ow we use the relation of orthogonality to define symmetric axis fol-lowing [Wu94, page 75]. For any pair A, B of two distinct points, let theunique line through the midpoint of A and B and perpendicular to the line AB be the perpendicular bisector of A, B . Clearly, if AB is an isotropic line,then its perpendicular bisector is AB itself.Let the perpendicular bisector of A, B be ℓ . We call A the symmetricpoint of B with respect to ℓ or ℓ the symmetric axis of A, B . Any point A on ℓ is said to be a symmetric point of itself with respect to ℓ . (Whenever ℓ is isotropic, it is the symmetric axis for any two points A, B on ℓ .) Wedenote this relation as Sym ( A, ℓ, B ). It will be important in our treatmentof Origami geometry.Let ℓ be any line and ℓ be a non-isotropic line. By [Wu94, page 76,Property 1], the points symmetric to the points from ℓ with respect to ℓ are also lying oa a uniquen line, say ℓ . In this case we call ℓ the symmetricline of ℓ with respect to ℓ , or ℓ the symmetric axis of ℓ and ℓ .Finally, we can add the following axiom to the list. (AxSymAx): Any two intersecting non-isotropic lines have a symmetricaxis.Now we are ready to give
Definition 5. A τ wu structure Π is a metric Wu plane if it is an orthogonalWu plane satisfying (AxSymAx). We would like to describe the correspondence between metric Wu planesand Pythagorean fields. Given a field F one can define a plane Π in astandard manner via Cartesian coordinates. Following the exposition ofMakowsky [Mak19], we denote such a Π as P P ∗ wu ( F ). On the other hand,given a plane Π we can use planary ternary rings to define a field F in thefollowing way introduced by M. Hall in [Hal43]. M. Hall credits [vS57, Hil71]for the original idea. A good exposition can be found in [Blu80, Szm83].We follow here almost verbatim [Iva16], which contains a particularly niceexposition of this construction.Let Π be a stucture satisfying (I-1, I-2, I-3) and (ParAx), with twodistinguished intersecting lines ℓ , m in Π. Let O be the point of intersectionof the lines ℓ and m . Take any point Z such that Z / ∈ ℓ ∪ m . Let d bethe line going through the points O and Z . Also let 1 be the point of the8ntersection of ℓ and a line m parallel to m and going through Z . Theconstruction will depend on a particular choice of parameters ℓ , m , and Z in the above configuration. Lemma 1.
There is a formula bij ( X, Y, ℓ , m , Z ) ∈ FOL ∈ which, for everychoice of ℓ , m and Z as above, defines a bijection between the points of ℓ and of m .Proof. Let X ∈ ℓ and h ( X ) be the point at the intersection of d of theline m parallel to m containing X . Let y ( X ) ∈ m be the point at theintersection of m of the line ℓ parallel to ℓ and containing h ( X ). Clearly f : ℓ → m given by f ( X ) = y ( X ) is a bijection and is FOL definable by aformula bij ( X, Y, ℓ , m , Z ).We will define a structure RF ∗ field (Π) whose universe will be denoted K (in fact, we take K = { A : A ∈ ℓ } ). Thinking of ℓ and m as the axisof a coordinate system we can identify the points of Π with pairs of pointsin K . The projection of a point P onto ℓ is defined by the point X ∈ ℓ which is the intersection of the line m parallel to m with P ∈ m . Afteranalogously projecting a point P onto m , we also need to use f in order toobtain a point in K . The point O has coordinates (0 , hascoordinates (1 , K will also be denoted bylower case letters. It should be clear from the context whether lower caseletters denote lines or elements of K .Next we define the slope sl ( ℓ ) ∈ K ∪ {∞} of a line ℓ in Π. If ℓ is parallelto ℓ , its slope is 0 and it is called a horizontal line. If ℓ is parallel to m ,its slope is ∞ and it is called a vertical line. For ℓ not vertical, let ℓ be theline parallel to ℓ and passing through 0. Let (1 , a ) be the coordinates of theintersection of ℓ with the line vertical line ℓ passing through (1 , sl ( ℓ ) = A ∈ K where A is a point in ℓ .This shows: Lemma 2.
There is a first order formula slope ( ℓ, A, ℓ , m , Z ) ∈ FOL ∈ which expresses sl ( ℓ ) = A , for any choice of parameters ℓ , m , Z . There isalso a first order formula slope ∞ ( ℓ, m ) ∈ FOL ∈ expressing sl ( ℓ ) = ∞ . Lemma 3. (i) Two lines ℓ, ℓ have the same slope, sl ( ℓ ) = sl ( ℓ ) iff theyare parallel.(ii) For the line d we have sl ( d ) = 1 (because (1 , ∈ d ).
9e now define a ternary operation T : K → K on the set K = { A : A ∈ ℓ } . We think of T ( a, x, b ) = h ax + b i as the result of multiplying a with x and then adding b . But we yet have to define multiplication and addition.Let a, b, x ∈ K . Let ℓ be the unique line with sl ( ℓ ) = a = ∞ intersectingthe line m at the point P with coordinates P = (0 , b ). Let ℓ = { ( x, z ) ∈ K : z ∈ K } . For every x ∈ K the line ℓ intersects ℓ at a unique point, say P = ( x, y ). We set T ( a, x, b ) = y . Lemma 4.
There is a formula Ter ( a, x, b, y, ℓ , m , Z ) ∈ FOL ∈ , where a, b, x, y range over coordinates and ℓ , m , Z are parameters of lines andpoints, which expresses that T ( a, x, b ) = y . Lemma 5.
The ternary operation T ( a, x, b ) has the following properties andinterpretations: (T-1): T (1 , x,
0) = T ( x, ,
0) = xT (1 , x,
0) = x means that the auxiliary line d = { ( x, x ) ∈ K : x ∈ K } is a line with sl ( d ) = 1. T ( x, ,
0) = x means that the slope of the line ℓ passing through (0 , , x ) is given by sl ( ℓ ) = x . (T-2): T ( a, , b ) = T (0 , a, b ) = b The equation T ( a, , b ) = b means that the line ℓ defined by T ( a, x, b ) = y intersects m at (0 , b ) (which is the meaning of ax + b in analyticgeometry).The equation T (0 , a, b ) = b means that the horizontal line ℓ passingthrough (0 , b ) consists of the points { ( a, b ) ∈ K : a ∈ K } . (T-3): For all a, x, y ∈ K there is a unique b ∈ K such that T ( a, x, b ) = y This means that for every slope s different from ∞ there is a uniqueline ℓ with sl ( ℓ ) = s passing through ( x, y ). (T-4): For every a, a ′ , b, b ′ ∈ K and a = a ′ the equation T ( a, x, b ) = T ( a ′ x, b ′ ) has a unique solution x ∈ K . This means that two lines ℓ and ℓ with different slopes not equal to ∞ intersect at a unique point P . (T-5): For every x, y, x ′ , y ′ ∈ K and x = x ′ there is a unique pair a, b ∈ K such that T ( a, x, b ) = y and T ( a, x ′ , b ) = y ′ . This means that any two points P , P not on the same vertical lineare contained in a unique line ℓ with slope different from ∞ .10 structure h K, T K , , i with a ternary operation T K and 0 , ∈ K satisfying (T-1)–(T-5) is called a planar ternary ring PTR. We also defineaddition add T ( a, b, c ) by T ( a, , b ) = c and multiplication mult T ( a, x, c ) by T ( a, x,
0) = c .Following [Mak19], we denote the structure ( K ; add T , mult T , ,
1) as RF ∗ field (Π). It is shown in [Hil13], that if Π is a Desarguesian plane, then RF ∗ field (Π) is a skew-field (a field, in which the commutativity of the multi-plication is not assumed) of characteristic 0. Moreover, as proved in [Wu94,page 42, Theorem 1], for any such Π and any two choices of ℓ , m , Z , thereis an isomorphism between the two obtained skew-fields.The following theorem is stated in this form in [Mak19], however it isbased on the previous classical results, in particular, by Hall [Hal43] andWu [Wu94]. Theorem 3. (i) Let F be a Pythagorean field of characteristic . Then P P ∗ wu ( F ) is a metric Wu plane.(ii) Let Π be a metric Wu plane. Then RF ∗ field (Π) is a Pythagorean fieldof characteristic .(iii) RF ∗ field ( P P ∗ wu ( F )) is isomorphic to F .(iv) P P ∗ wu ( RF ∗ field (Π)) is isomorphic to Π .Proof. (i) Axioms (I-1, I-2, I-3) and (ParAx) are shown in [Har00, Propo-sition 14.1]. The infinity axiom holds, since F has characteristic 0. Con-sidering Desargues’ axioms, Proposition 14.4 in [Har00] shows that Pappustheorem holds in a plane defined over a field. Then by Hessenberg’s theo-rem [Wu94, page 67], Desargues’ axioms also hold.We naturally define lines a x + b y + c = 0 and a x + b y + c = 0 to beorthogonal if a a + b b = 0. Then the axiom (O-1) holds by commutativityof multiplication. The axioms (O-2), (O-3) and (O-5) hold since we are ableto solve systems of linear equations. We know that 1 = 0 and hence anyline of the form x = c is non-isotropic. Then for any point ( x , y ) there isa non-isotropic line x = x passing through it and (O-4) holds.Suppose an angle is formed by two non-isotropic lines given by l x + m y + n = 0 and l x + m y + n = 0. Then the internal and externalbisectors are given by the two equations l x + m y + n p l + m = ± l x + m y + n p l + m . F is Pythagorean, the roots exist and are not equal to 0, because thelines are non-isotropic. Therefore, bisectors exist and (AxSymAx) holds.(ii) This statement is extensively discussed in [Wu94]. As mentionedabove, Hilbert showed in Grundlagen der Geometrie that RF ∗ field (Π) formsa skew-field of characteristic 0. Wu first proves in [Wu94, Section 2.1] thatLinear Pascalian axiom (a version of Pappus theorem) is sufficient to obtainthe commutativity of multiplication and then on [Wu94, page 72] shows thatLinear Pascalian axiom holds in any metric Wu plane.Finally, to conclude that RF ∗ field (Π) is Pythagorean, we refer to the Kou-Ku Theorem (in the Western tradition known as Pythagorean Theorem)proved on [Wu94, page 97].(iii) As mentioned above, it is shown in [Wu94, page 42, Theorem 1],that a different choice of parameters in the construction of RF ∗ field (Π) givesus isomorphic fields. Clearly, if we choose ℓ , m to be the axes of P P ∗ wu ( F )and Z = (1 , F . Hence,it will be isomorphic to RF ∗ field ( P P ∗ wu ( F )) for every choice of ℓ , m , Z .This result can also be found in [Hal43, Theorem 5.9].(iv) Let Π be an orthogonal plane and let F = RF ∗ field (Π). In order toestablish an isomorphism between P P ∗ wu ( F ) and Π we need to define twomaps: a map of points and a map of lines.Points of P P ∗ wu ( F ) are pairs ( x, y ) ∈ F . We recall that the universe of F is the set of points incident to the line ℓ , and that there is a definablebijection f between the points of ℓ and m (the coordinate axes in Π,parameters of the considered interpretation). Hence, given ( x, y ) we candefine two auxiliary lines: m x going through x ∈ ℓ and parallel to m , and ℓ y , going through f ( y ) ∈ m and parallel to ℓ . Let A be the intersectionof m x and ℓ y . We map ( x, y ) to A , and it is clear that A has coordinates( x, y ). Thus, we have described a (definable) bijection between the sets ofpoints of P P ∗ wu ( F ) and Π.Lines of P P ∗ wu ( F ) can be specified by equations ax + by + c = 0 wherenot all of a, b, c are 0. Thus, a line is interpreted by a triple ( a, b, c ) ∈ F .Two lines are defined to be equal if ( a, b, c ) and ( a ′ , b ′ , c ′ ) are proportional.A point ( x, y ) is incident to a line ( a, b, c ) if ax + by + c = 0.Each line in P P ∗ wu ( F ) is equal to a line defined by the equation y = ax + b or to a vertical line x = c . We construct the corresponding line in Π bydrawing, in the first case, a line through the point (0 , b ) with the slope a ,and in the second case a vertical line (parallel to m ) through ( c, P P ∗ wu ( F ) to lines in Π and is clearly a (definable) bijectionpreserving the incidence relation.Concerning the orthogonality relation, we may assume that the coor-12inate axes ℓ and m in Π are selected to be orthogonal. (By Wu, thefield F does not depend on the choice of parameters, up to isomorphism.)We can define lines ( a, b, c ) and ( a ′ , b ′ , c ′ ) in P P ∗ wu ( F ) to be orthogonal iff aa ′ + bb ′ = 0. Then the usual arguments show that this agrees with theorthogonality of the corresponding lines in Π. Remark 1.
Theorem 3 states that the theories of orthogonal Wu planesand of Pythagoren fields are bi-interpretable. Albert Visser considered astronger notion of bi-interpretability that, in addition, requires that theisomorphisms in items (iii) and (iv) be internally definable, respectively, in F and Π. It is clear from the given proof that in our situation this is, indeed,the case. To establish the undecidability of the theory of Pythagorean fields we referto the following theorem of M. Ziegler [Zie82, Bee]:
Theorem 4.
Let T be a finite subtheory of the theory of the field of reals ( R ; + , × ) . Then(i) T is undecidable;(ii) The same holds for the extension of T by the axioms stating that thecharacteristic of the field is . Although the second part is not mentioned as a result in [Zie82, Bee],it easily follows from Ziegler’s proof.
Corollary 1.
The theories of Pythagorean fields and of Pythagorean fieldsof characteristic are undecidable. Using the bi-interpretability of Pythagorean fields and metric Wu planeswe obtain the undecidability of the theory of metric Wu planes. In fact,we prove a more general theorem establishing the undecidability of a suffi-ciently wide class of geometric theories. As a preparation for its proof, wedefine syntactic translations between formulas in the language of fields andformulas in the language τ wu .Consider any formula φ in the language of fields. Using the formulas add and mult from the construction of RF ∗ field (Π) to interpret addition and13ultiplication, we obtain a formula φ wu ( ℓ , m , Z ) in the language τ wu ,where ℓ , m , Z are the parameters (free variables) of the formula.Let Par ( ℓ , m , Z ) denote the formula stating that ℓ and m are linesintersecting in exactly one point and that Z is not incident with either ℓ or m . These conditions definably specify the admissible values of theparameters. Then, for any metric Wu plane Π, RF ∗ field (Π) | = φ ⇐⇒ Π | = ∀ ℓ , m , Z ( Par ( ℓ , m , Z ) → φ wu ) . (1)Similarly, for the other interpretation, consider any formula φ in thelanguage of τ wu . Using the formulas from the construction of P P ∗ wu tointerpret the two sorts of variables, equality, incidence and orthogonality,we obtain a formula φ field in the language of fields. Then, for any field F , P P ∗ wu ( F ) | = φ ⇐⇒ F | = φ field . (2)Now we are ready to state a geometric version of Ziegler’s Theorem.Let Π R denote the real plane P P ∗ wu ( R ). Let WU denote the first ordertheory of metric Wu planes and let PF denote the first order theory ofPythagorean fields of characteristic 0. Theorem 5.
Let T be a finite set of axioms in the vocabulary τ wu such that Π R | = T . Then T ∪ WU is undecidable.Proof. Let T ′ = { φ field | φ ∈ T } . Then by Ziegler’s theorem, T ′ ∪ PF isundecidable.We want to prove that T ′ ∪ PF | = φ ⇐⇒ T ∪ WU | = ∀ ℓ , m , Z ( Par ( ℓ , m , Z ) → φ wu ) . (3)Then, since the translation ( · ) wu is computable, this provides a computablereduction of T ′ ∪ PF to T ∪ WU and proves that the latter is undecidable.To prove (3), suppose T ∪ WU | = ∀ ℓ , m , Z ( Par ( ℓ , m , Z ) → φ wu ).Take any F | = T ′ ∪ PF. Then by Theorem 3,
P P ∗ wu ( F ) is a metric Wu planeand, using (2), we have P P ∗ wu ( F ) | = T ∪ WU. Hence,
P P ∗ wu ( F ) | = ∀ ℓ , m , Z ( Par ( ℓ , m , Z ) → φ wu ) . Then, by (1), RF ∗ field ( P P ∗ wu ( F )) | = φ and, since F ∼ = RF ∗ field ( P P ∗ wu ( F )), weobtain T ′ ∪ PF | = φ .Suppose T ′ ∪ PF | = φ . Consider any Π | = T ∪ WU. Let F = RF ∗ field (Π).By Theorem 3, F is a Pythagorean field of characteristic 0 and Π is iso-morphic to P P ∗ wu ( F ). Then, using (2), we obtain F | = T ′ ∪ PF. Therefore,14 | = φ and by (1), Π | = ∀ ℓ , m , Z ( Par ( ℓ , m , Z ) → φ wu ) . It follows that T ∪ WU | = ∀ ℓ , m , Z ( Par ( ℓ , m , Z ) → φ wu ).This completes the proof of (3) and thereby of Theorem 5. Corollary 2.
The theory of metric Wu planes is undecidable.
Huzita–Justin axioms were not meant to be axioms in the logical sense, butrather rules of folding. Yet, one can try to naively formulate them in thelanguage τ wu by treating the requirement to construct an object (satisfyinggiven conditions) by a classical existential statement. Huzita–Justin axiomsare naturally stated using the relation Sym ( P , ℓ, P ) “points P and P aresymmetric with respect to line ℓ ” defined in Section 3. Then one obtains thefollowing versions of Huzita–Justin axioms. (H-1): Given two points P and P , construct a unique fold (line) thatpasses through both of them: ∀ P , P ∃ =1 ℓ ( P ∈ ℓ ∧ P ∈ ℓ ) . (H-2): Given two points P and P , construct a unique fold (line) thatplaces P onto P : ∀ P , P ∃ =1 ℓ Sym ( P , ℓ, P ) . (H-3): Given two lines ℓ and ℓ , construct a fold (line) that places ℓ onto ℓ : ∀ ℓ , ℓ ∃ k ∀ P ( P ∈ ℓ → ∃ P ( P ∈ ℓ ∧ Sym ( P , k, P ))) . (H-4): Given a point P and a line ℓ , construct a unique fold (line) orthog-onal to ℓ that passes through P : ∀ P, ℓ ∃ =1 k ( P ∈ k ∧ ℓ ⊥ k ) . (H-5): Given two points P and P and a line ℓ , construct a fold (line)that places P onto ℓ and passes through P : ∀ P , P , ℓ ∃ ℓ ( P ∈ ℓ ∧ ∃ P ( Sym ( P , ℓ , P ) ∧ P ∈ ℓ )) . H-6):
Given two points P and P and two lines ℓ and ℓ , construct afold (line) that places P onto ℓ and P onto ℓ : ∀ P , P , ℓ , ℓ ∃ ℓ ( ∃ Q ( Sym ( P , ℓ , Q ) ∧ Q ∈ ℓ ) ∧∧∃ Q ( Sym ( P , ℓ , Q ) ∧ Q ∈ ℓ )) . Since the original Huzita-Justin axioms talk only about the possibility ofthe existence of folds, Axioms (H-5) and (H-6) formulated above do not holdin a real plane. The exceptional configurations, where a described fold doesnot exist have to be described explicitly. In order to fix that we are going toamend them with the most obvious conditions under which the lines wouldindeed exist. On the other hand, the formalizations of Axioms (H-1, H-2,H-3, H-4) obviously hold in the real plane and, as explained below, almosthold in any metric Wu plane.Following [Alp00], we would like to state that metric Wu planes satisfythe first four Origami axioms (H-1, H-2, H-3, H-4). However, since therecould exist non-isotropic lines in a metric Wu plane, they do not necessarilysatisfy (H-3). Thus, we modify this axiom.
Definition 6. (H*3) Given two non-isotropic lines ℓ and ℓ , there is afold (line) that places ℓ onto ℓ . Proposition 1.
Every metric Wu plane satisfies the Origami axioms (H-1,H-2, H*3, H-4).
Proof. (H-1) is equivalent to (I-1) and (H-4) is equivalent to (O-2). To prove(H-2) we use the construction from [Wu94, page 75].(AxSymAx) is an analogue of (H*3) for intersecting lines, so we mayonly consider the case of parallel lines. Take ℓ and ℓ parallel to each other.Take any point P ∈ ℓ and drop a perpendicular from P on ℓ . Let theintersecting point be P . Then we claim that the perpendicular bisector of P P is the line we need. Inspired by [Alp00] we want to establish a correspondence between Euclideanfields and planes satisfying some analogue of (H-5). Euclidean fields areordered, therefore we want our plane to be in some sense “ordered” as well.One way to do so would be to take as an axiom that there are no isotropiclines. Then the corresponding field would be formally real and hence order-able (see below). We take a different approach and following [Wu94] intro-duce a new relation of Betweenness Be ( P , P , P ) to be interpreted as three istinct points are on the same line and P is between P and P . Let τ o − wu be the signature consisting of ∈ , ⊥ and Be . Axioms of betweenness(B-1):
Let
A, B, C be three distinct points on a line. If B lies between A and C , then B also lies between C and A . (B-2): For any two distinct points A and C on a line, there always existsanother point B which lies between A and C , and another point D such that C lies between A and D . (B-3): Given any three distinct points
A, B, C on a line, one and only oneof the following three cases holds: B lies between A and C , A liesbetween B and C , and C lies between A and B . (B-4): (Pasch) Assume the points A, B, C and ℓ in general position, i.e.the three points are not on one line, none of the points is on ℓ . Let D be the point at which ℓ and the line AB intersect. If Be ( A, D, B )there is D ′ ∈ ℓ with Be ( A, D ′ , C ) or Be ( B, D ′ , C ). Definition 7. A τ o − wu structure Π is an ordered metric Wu plane if it is ametric Wu plane satisfying axioms of betweenness (B-1, B-2, B-3, B-4). Proposition 2.
Every ordered metric Wu plane satisfies (H-3) and hencethe Origami axioms (H-1, H-2, H-3, H-4).
Proof.
There are no isotropic lines in ordered metric Wu planes as provenin [Wu94, page 107, Theorem 3].If F is an ordered field, we define the relation of betweenness on P P ∗ wu ( F )in the standard way. If Π is an ordered metric Wu plane, we follow [Wu94,page 105] to define an order on RF ∗ field (Π). By [Wu94, page 103, Separationproperty 1], all points on ℓ distinct from 0 can be separated into two parts,such that 0 lies between A, B when
A, B lie on different sides, and 0 does notlie between
A, B when
A, B lie on the same side. We define those numbersin RF ∗ field (Π) whose corresponding points lie on the same side of 0 on ℓ as1 to be positive numbers and those whose corresponding points lie on theother side of 0 on ℓ to be negative numbers. Then we can say that a < b whenever b − a is a positive number. Theorem 6. (i) Let F be an ordered Pythagorean field. Then P P ∗ wu ( F ) is an ordered metric Wu plane. ii) Let Π be an ordered metric Wu plane. Then RF ∗ field (Π) is an orderedPythagorean field.(iii) RF ∗ field ( P P ∗ wu ( F )) is isomorphic to F .(iv) P P ∗ wu ( RF ∗ field (Π)) is isomorphic to Π .Proof. (i) Using properties of ordered fields, it is easy to check that P P ∗ wu ( F )satisfies the axioms of betweenness (B-1, B-2, B-3, B-4).(ii) We only need to show that RF ∗ field (Π) is an ordered field. This isproved on [Wu94, page 105, Theorem 1].(iii) By Theorem 3 it is sufficient to check that the relation of order ispreserved. As discussed on [Wu94, page 105], if we take different ℓ , m , Z in the construction of RF ∗ field (Π), the canonical isomorphism betweenthe obtained fields will preserve order. If we choose ℓ , m to be the axes of P P ∗ wu ( F ) and Z = (1 , F . It followsthat F is isomorphic to RF ∗ field ( P P ∗ wu ( F )) for any choice of parameters.(iv) Let F denote the field RF ∗ field (Π). By Theorem 3 it is sufficient tocheck that the betweenness relation is preserved under the isomorphism ofmetric Wu planes Π and P P ∗ wu ( F ).Since the collinearity is preserved, it is sufficient to consider the between-ness relation for points on the same line. Suppose three points A, B, C on aline ℓ in Π are given. Assume ℓ is not vertical and consider their coordinateprojections on ℓ axis. By Corollary 1 in [Wu94, page 104] we know that thebetweenness relation is preserved by parallel projection. On the other hand,the interpretation of betweenness in P P ∗ wu ( F ) for points of the coordinateaxis is the same as that in Π. This shows the claim in the case ℓ is notvertical.If ℓ is vertical, we consider the projections of A, B, C on the m axis andthe corresponding points on ℓ via the bijection f . By the same principle, f preserves the betweenness on respective coordinate axes, hence Be ( A, B, C )holds in Π iff it holds in
P P ∗ wu ( F ).After establishing Theorem 6, we also obtain an ordered analogue ofTheorem 5. Although in [Zie82, Bee] only theories in the language of fieldsare concerned, the proof still holds for the case of ordered fields, which givesus the following result. Theorem 7.
Let T be a finite subtheory of the theory of the ordered field ofreals ( R ; + , × , ≤ ) . Then(i) T is undecidable; ii) The same holds for the extension of T by the axioms stating that thecharacteristic of the field is . Then using Theorem 6 and the same technique as in Theorem 5, weobtain an ordered version of the geometrical Ziegler’s theorem.Let Π R denote the real plane P P ∗ wu ( R ). Let OWU denote the first ordertheory of ordered metric Wu planes. Theorem 8.
Let T be a finite set of axioms in the vocabulary τ o − wu suchthat Π R | = T . Then T ∪ OWU is undecidable.
Next we consider the question of orderability. Recall that a field isorderable (or formally real ) if − − Proposition 3.
A metric Wu plane Π is orderable iff there are no isotropiclines in Π.
Proof.
In one direction, we have already mentioned a theorem of Wu thatordered metric Wu planes have no isotropic lines. In the other direction, weassume a Π without isotropic lines is given and consider as a coordinate sys-tem a pair of orthogonal lines and the corresponding field F = RF ∗ field (Π).By Theorem 3 (iv) P P ∗ wu ( F ) is isomorphic to Π.We claim that F is formally real. Assume otherwise, then d + 1 = 0in F , for some d . Then the line defined by the points (1 ,
0) and (0 , d ) (andthe parallel line given by the equation dx + y = 0) is isotropic. Since F isformally real, P P ∗ wu ( F ) is orderable, but it is isomorphic to Π. Following [Alp00], we want to establish a correspondence between Euclideanfields, defined below, and planes satisfying some amended version of (H-5).
Definition 8. A Euclidean field is a formally real Pythagorean field suchthat every element is either a square or the opposite of a square: ∀ x ∃ y ( x = y ∨ − x = y ) . The nonzero squares of a Euclidean field constitute a positive cone, hence(see [Bec74]) Euclidean fields admit a unique ordering: x ≤ y ⇐⇒ ∃ z ( x + z = y ) . Proposition 4.
The first order theory of Euclidean fields is undecidable.
Proof.
Ziegler’s Theorem.Next we formulate our amended version of Axiom (H-5) in which we addan appropriate precondition for the constructed fold to exist. Below we usethe notion a point A is closer to a line ℓ than to a point B , Closer ( A, ℓ, B ),which can be formulated in the language τ o − wu by saying that there ex-ist points H ∈ ℓ and B ′ and a line m ∋ A such that Sym ( H, m, B ′ ) and Be ( A, B ′ , B ). Definition 9. (H*5) Given two points P and P and a line ℓ , if P iscloser to ℓ than to P , then there is a fold (line) that places P onto ℓ andpasses through P . Definition 10. A τ o − wu structure Π is a Euclidean ordered metric Wu plane if it is an ordered metric Wu plane satisfying (H*5).
Theorem 9. (i) Let F be a Euclidean field. Then P P ∗ wu ( F ) is a Eu-clidean ordered metric Wu plane.(ii) Let Π be a Euclidean ordered metric Wu plane. Then RF ∗ field (Π) is aEuclidean field.(iii) Furthermore, RF ∗ field ( P P ∗ wu ( F )) is isomorphic to F .(iv) P P ∗ wu ( RF ∗ field (Π)) is isomorphic to Π .Proof. (i) In order to show (H*5) it is enough to prove that a circle inter-sects a line whenever the radius is smaller than the distance betweenthe center and the line. In P P ∗ wu ( F ) that is equivalent to solving aquadratic equation. Therefore, if F is Euclidean, (H*5) holds.(ii) Consider any positive s ∈ RF ∗ field (Π) and show that the square root of s exists. Without loss of generality we assume s >
1, since otherwisewe can find a square root of s − . Let P = (0 , s ), P = (0 , s − ) andlet ℓ be the x axis. Then by (H*5) we can find a point P = ( x, y ),such that P ∈ ℓ and | P P | = | P P | . That means y = 0 and (cid:0) s − (cid:1) + x = (cid:0) s +12 (cid:1) , giving x = s .20iii), (iv) We use Theorem 6, since every Euclidean field is a Pythagorean fieldand every Euclidean ordered metric Wu plane is an ordered metric Wuplane.Then by Theorem 8, we obtain: Corollary 3.
The theory of Euclidean ordered Wu planes is undecidable.
Finally, we would like to find a correct version of (H*6) and to establishits correspondence with Vieta fields as defined below.
Definition 11. A Vieta field is a Euclidean field in which every element isa cube: ∀ x ∃ y y = x. It follows from Cardano formula that any cubic polynomial over a Vietafield has at least one root.
Proposition 5.
The first order theory of Vieta fields is undecidable.
Proof.
Ziegler’s Theorem.The following version of (H-6) is inspired by [GKK12, Proposition 6].
Definition 12. (H*6) Given two points P and P and two lines ℓ and ℓ , if P / ∈ ℓ , P / ∈ ℓ , ℓ and ℓ are not parallel and points are distinct orlines are distinct, then there is a fold (line) that places P onto ℓ and P onto ℓ . Definition 13.
Π is an ordered Wu Origami plane if it is an ordered metricWu plane which also satisfies (H*5) and (H*6).
Theorem 10. (i) Let F be a Vieta field. Then P P ∗ wu ( F ) is an orderedWu Origami plane.(ii) Let Π be an ordered Wu Origami plane. Then RF ∗ field (Π) is a Vietafield.(iii) Furthermore, RF ∗ field ( P P ∗ wu ( F )) is isomorphic to F .(iv) P P ∗ wu ( RF ∗ field (Π)) is isomorphic to Π . roof. (i) It sufficies to show that (H*6) holds in P P ∗ wu ( F ). We use[GKK12, Proposition 6] to conclude that the conditions we chose aresufficient for the existence of a fold. Although the original result wasproven specifically for the real plane, we note that it essentially usesonly the Vieta property of R and therefore holds for any plane over aVieta field.(ii) Take any r ∈ RF ∗ field (Π). Let P = ( − , P = (0 , − r ), ℓ be theline x = 1 and ℓ the line y = r . Then by (H*6) we can find a line ℓ .Then if one drops a perpendicular from P on ℓ , the constructed pointwill have coordinates (0 , s ), where s = r .(iii), (iv) Once again this follows from Theorem 6, since every Vieta field is aPythagorean field and every ordered Wu Origami plane is an orderedmetric Wu plane. Corollary 4.
The theory of ordered Wu Origami planes is undecidable.
We have axiomatized the classes of orthogonal Wu planes using versions ofOrigami axioms and established their bi-interpretations with the first ordertheories of fields (corresponding to the classes of Pythagorean, Euclideanand Vieta fields). A few natural questions concerning the axiomatization ofgeometry via Origami constructions were left open.One such question concerns the choice of the considered language. Al-though the orthogonality of lines is natural from the point of view of Origamiconstructions — orthogonality can be tested just by a single fold — we seethat the Huzita–Justin axioms are easily formulated using the notion ofsymmetry of two points w.r.t. a line
Sym ( P, ℓ, Q ). It would be natural toconsider this notion as basic and orthogonality as definable. The
Sym pred-icate behaves well provided the metric Wu plane is orderable, that is, hasno isotropic lines.
Problem 1.
Find a natural axiomatization of orderable metric Wu planesin terms of ∈ and Sym .A similar question can be asked about betweenness. We have based ouraxiomatization on the standard Hilbertian axioms for betweenness. Onecan, however, consider as basic the relation
Closer ( A, ℓ, B ) which holds if A
22s closer to line ℓ than to B . It is the relation that was used in the statementof (H*5). Problem 2.
Find a natural axiomatization of the class of ordered metricWu planes in terms of ∈ , ⊥ and Closer . In particular, this requires thatthere is a first-order formula that works as a definition of betweenness ineach structure satisfying these axioms.Another question concerns the definability of betweenness in Euclideanmetric Wu planes. Recall that in Euclidean fields the ordering is definable.This suggests that there is an axiomatization of Euclidean ordered metricWu planes in the language τ wu only. In fact, one such axiomatization basedon the so-called Euclidean axiom of betweenness is well-known [Wika]. Thisaxiom defines the betweenness relation Be ( A, B, C ) by stating that
A, B, C are collinear and there exists a point D such that DA ⊥ DC and DB ⊥ AC .Then, a metric Wu plane is Euclidean ordered iff it has no isotropic linesand the above relation Be satisfies the usual axioms of betweenness.It would be interesting to know if the Euclidean axiom of betweenness canbe replaced by its alternative suggested by the Origami axiom (H-5). First, define Closer ( A, ℓ, B ) by saying that there is a fold m that goes through A and places B on ℓ . Second, define Be ( A, B, C ) by saying that
A, B, C are collinear and there is a line ℓ ∋ D such that ℓ ⊥ AC , Closer ( A, ℓ, C )and
Closer ( C, ℓ, A ). State (some of) the betweenness axioms for the definedrelation. Though this approach may work, there are a number of details tobe worked out here that we leave for a future study.
Problem 3.
Find an axiomatization of Euclidean orderable Wu planes inthe language τ wu that would be natural from the point of view of Origami.Yet another interesting direction of study is to develop a constructiveversion of Origami geometry as a logical theory based on intuitionistic logic,in the spirit of the work of Beeson [Bee15]. In such a theory existentialstatements would yield actual Origami constructions rather than just beclassically true. Problem 4.
Develop a constructive version of Origami geometry.
10 Acknowledgements
The work of Lev Beklemishev and Anna Dmitrieva was supported by theAcademic Fund Program at the National Research University Higher Schoolof Economics (HSE) in 2019–2020 (grant No. 19-04-050) and by the RussianAcademic Excellence Project “5–100”.23 eferences [Alp00] R.C. Alperin. A mathematical theory of origami constructionsand numbers.
New York J. Math , 6(119):133, 2000.[Bec74] Eberhard Becker. Euklidische K¨orper und euklidische H¨ullen vonK¨orpern.
Journal f¨ur die reine und angewandte Mathematik
An-nals of Pure and Applied Logic , 166(11):1199 – 1273, 2015.[Blu80] L. M. Blumenthal.
A modern view of geometry . Courier Corpo-ration, 1980.[GKK12] Fadoua Ghourabi, Asem Kasem, and Cezary Kaliszyk. Algebraicanalysis of Huzita’s origami operations and their extensions. In
International Workshop on Automated Deduction in Geometry ,pages 143–160. Springer, 2012.[Hal43] M. Hall. Projective planes.
Transactions of the American Math-ematical Society , 54(2):229–277, 1943.[Har00] R. Hartshorne.
Geometry: Euclid and Beyond . Springer, 2000.[Hil71] D. Hilbert.
Foundations of Geometry, Second Edition, translatedfrom the Tenth Edition, revised and enlarged by Dr Paul Bernays .The Open Court Publishing Company, La Salle, Illinois, 1971.[Hil13] D. Hilbert.
Grundlagen der Geometrie . Springer-Verlag, 2013.[Iva16] N.V. Ivanov. Affine planes, ternary rings, and examples of non-desarguesian planes. arXiv preprint arXiv:1604.04945 , 2016.[Jus89] J. Justin. R´esolution par le pliage de ´equation du troisieme degr´eet applications g´eom´etriques. In
Proceedings of the first interna-tional meeting of origami science and technology , pages 251–261.Ferrara, Italy, 1989. 24Mak18] Johann A. Makowsky. The undecidability of orthogonal andorigami geometries. In
International Workshop on Logic, Lan-guage, Information, and Computation , pages 250–270. Springer,2018.[Mak19] Johann A. Makowsky. Can one design a geometry engine?
Annalsof Mathematics and Artificial Intelligence , 85(2-4):259–291, 2019.[Szm83] W. Szmielew.
From affine to Euclidean geometry, an axiomaticapproach . Polish Scientific Publishers (Warszawa-Poland) and D.Reidel Publishing Company (Dordrecht-Holland), 1983.[vS57] K.G.C. von Staudt.
Beitr¨age zur Geometrie der Lage , volume 2.F. Korn, 1857.[Wika] Wikipedia. Euklidischer K¨orper. Wikipedia entry:https://de.wikipedia.org/wiki/Euklidischer K¨orper.[Wikb] Wikipedia. Huzita-Hatori axioms. Wikipedia entry:https://en.wikipedia.org/wiki/Huzita-Hatori axioms.[Wu94] W.-T. Wu.
Mechanical Theorem Proving in Geometries, Springer1994 . Springer, 1994. (Original in Chinese, 1984).[Zie82] Martin Ziegler. Einige unentscheidbare K¨orpertheorien. InV. Strassen E. Engeler, H. L¨auchli, editor,