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Journal articles on the topic 'Geometry Axioms'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Richter, William, Adam Grabowski, and Jesse Alama. "Tarski Geometry Axioms." Formalized Mathematics 22, no. 2 (June 30, 2014): 167–76. http://dx.doi.org/10.2478/forma-2014-0017.

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Summary This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski [8], and we hope to continue this work. The article is an extension and upgrading of the source code written by the first author with the help of miz3 tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert’s axioms. This is largely a Mizar port of Julien Narboux’s Coq pseudo-code [6]. We partially prove the theorem of [7] that Tarski’s (extremely weak!) plane geometry axioms imply Hilbert’s axioms. Specifically, we obtain Gupta’s amazing proof which implies Hilbert’s axiom I1 that two points determine a line. The primary Mizar coding was heavily influenced by [9] on axioms of incidence geometry. The original development was much improved using Mizar adjectives instead of predicates only, and to use this machinery in full extent, we have to construct some models of Tarski geometry. These are listed in the second section, together with appropriate registrations of clusters. Also models of Tarski’s geometry related to real planes were constructed.

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2

Coghetto, Roland, and Adam Grabowski. "Tarski Geometry Axioms – Part II." Formalized Mathematics 24, no. 2 (June 1, 2016): 157–66. http://dx.doi.org/10.1515/forma-2016-0012.

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Summary In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness identity (A6), Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8), the upper dimension axiom (A9), the Euclid axiom (A10), the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS). In order to show that the structure which satisfies all eleven Tarski’s axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes. Although the tradition of the mechanization of Tarski’s geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned – we had some doubts about the proof of the Euclid’s axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].

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3

Grigoryan, Yu. "Axioms of Heterogeneous Geometry." Cybernetics and Systems Analysis 55, no. 4 (July 2019): 539–46. http://dx.doi.org/10.1007/s10559-019-00162-3.

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4

BEESON, MICHAEL, PIERRE BOUTRY, and JULIEN NARBOUX. "HERBRAND’S THEOREM AND NON-EUCLIDEAN GEOMETRY." Bulletin of Symbolic Logic 21, no. 2 (June 2015): 111–22. http://dx.doi.org/10.1017/bsl.2015.6.

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AbstractWe use Herbrand’s theorem to give a new proof that Euclid’s parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.

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5

Coghetto, Roland, and Adam Grabowski. "Tarski Geometry Axioms. Part III." Formalized Mathematics 25, no. 4 (December 20, 2017): 289–313. http://dx.doi.org/10.1515/forma-2017-0028.

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Summary In the article, we continue the formalization of the work devoted to Tarski’s geometry - the book “Metamathematische Methoden in der Geometrie” by W. Schwabhäuser, W. Szmielew, and A. Tarski. After we prepared some introductory formal framework in our two previous Mizar articles, we focus on the regular translation of underlying items faithfully following the abovementioned book (our encoding covers first seven chapters). Our development utilizes also other formalization efforts of the same topic, e.g. Isabelle/HOL by Makarios, Metamath or even proof objects obtained directly from Prover9. In addition, using the native Mizar constructions (cluster registrations) the propositions (“Satz”) are reformulated under weaker conditions, i.e. by using fewer axioms or by proposing an alternative version that uses just another axioms (ex. Satz 2.1 or Satz 2.2).

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6

von Plato, Jan. "The axioms of constructive geometry." Annals of Pure and Applied Logic 76, no. 2 (December 1995): 169–200. http://dx.doi.org/10.1016/0168-0072(95)00005-2.

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7

Tarski, Alfred, and Steven Givant. "Tarski's System of Geometry." Bulletin of Symbolic Logic 5, no. 2 (June 1999): 175–214. http://dx.doi.org/10.2307/421089.

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AbstractThis paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

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8

Gottlieb, Alex D., and Joseph Lipman. "Group-Theoretic Axioms For Projective Geometry." Canadian Journal of Mathematics 43, no. 1 (February 1, 1991): 89–107. http://dx.doi.org/10.4153/cjm-1991-006-2.

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AbstractWe show that a certain category 𝓖 whose objects are pairs G ⊃ H of groups subject to simple axioms is equivalent to the category of ≧ 2-dimensional vector spaces and injective semi-linear maps; and deduce via the "Fundamental Theorem of Projective Geometry" that the category of ≧ 2-dimensional projective spaces is equivalent to the quotient of a suitable subcategory of 𝓖 by the least equivalence relation which identifies conjugation by any element of H with the identity automorphism of G.

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9

Coghetto, Roland, and Adam Grabowski. "Tarski Geometry Axioms. Part IV – Right Angle." Formalized Mathematics 27, no. 1 (April 1, 2019): 75–85. http://dx.doi.org/10.2478/forma-2019-0008.

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Summary In the article, we continue [7] the formalization of the work devoted to Tarski’s geometry – the book “Metamathematische Methoden in der Geometrie” (SST for short) by W. Schwabhäuser, W. Szmielew, and A. Tarski [14], [9], [10]. We use the Mizar system to systematically formalize Chapter 8 of the SST book. We define the notion of right angle and prove some of its basic properties, a theory of intersecting lines (including orthogonality). Using the notion of perpendicular foot, we prove the existence of the midpoint (Satz 8.22), which will be used in the form of the Mizar functor (as the uniqueness can be easily shown) in Chapter 10. In the last section we give some lemmas proven by means of Otter during Tarski Formalization Project by M. Beeson (the so-called Section 8A of SST).

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10

BEESON, MICHAEL. "CONSTRUCTIVE GEOMETRY AND THE PARALLEL POSTULATE." Bulletin of Symbolic Logic 22, no. 1 (March 2016): 1–104. http://dx.doi.org/10.1017/bsl.2015.41.

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AbstractEuclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom (dating from 1795), which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions.

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11

Dawkins, Paul Christian. "Student interpretations of axioms in planar geometry." Investigations in Mathematics Learning 10, no. 4 (January 24, 2018): 227–39. http://dx.doi.org/10.1080/19477503.2017.1414981.

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12

Petrov, A. P., and L. V. Kuzmin. "Visual space geometry derived from occlusion axioms." Journal of Mathematical Imaging and Vision 6, no. 2-3 (June 1996): 291–308. http://dx.doi.org/10.1007/bf00119844.

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13

Schutz, John W. "A System of Axioms for Hyperbolic Geometry." Journal of Geometry 90, no. 1-2 (November 21, 2008): 185–92. http://dx.doi.org/10.1007/s00022-008-1903-9.

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14

Hart, Joan E., and Kenneth Kunen. "Weak measure extension axioms." Topology and its Applications 85, no. 1-3 (May 1998): 219–46. http://dx.doi.org/10.1016/s0166-8641(97)00147-8.

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15

Jeldtoft Jensen, Henrik, and Piergiulio Tempesta. "Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory." Entropy 20, no. 10 (October 19, 2018): 804. http://dx.doi.org/10.3390/e20100804.

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The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed degrees of freedom N. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume W on N. We review the ensuing entropies, discuss the composability axiom and explain why group entropies may be particularly relevant from an information-theoretical perspective.

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16

Cullinane, Michael J. "Metric axioms and distance." Mathematical Gazette 95, no. 534 (November 2011): 414–19. http://dx.doi.org/10.1017/s0025557200003508.

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Many mathematicians' experiences with distances in the settings of geometry, analysis, and topology can lead to the impression that the only worthwhile or ‘reasonable’ distance functions are metrics. We hope to convince the reader otherwise.Recall that a metric for a set X is a function d: X × X → [0, ∞) satisfying all of the following metric axioms:

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17

Fehr, Howard F. "The Present Year-Long Course in Euclidean Geometry Must Go." Mathematics Teacher 100, no. 3 (October 2006): 165–68. http://dx.doi.org/10.5951/mt.100.3.0165.

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It is assumed that the geometry course refers to one that is commonly taught in the tenth school year. It is traditional Euclidean synthetic geometry, of 2- and 3-space, modified by an introduction of ruler and protractor axioms into the usual synthetic axioms. A unit of coordinate geometry of the plane is usually appended. It is a course that is reflected in textbooks prepared by the School Mathematics Study Group and in most commercial textbooks.

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18

RENNIE, A. "COMMUTATIVE GEOMETRIES ARE SPIN MANIFOLDS." Reviews in Mathematical Physics 13, no. 04 (April 2001): 409–64. http://dx.doi.org/10.1142/s0129055x01000673.

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In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spinc geometry depending on whether the geometry is "real" or not. We attempt to flesh out the details of Connes' ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemannian spin manifolds. Throughout we are as explicit and elementary as possible.

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Fehr, Howard F. "The Present Year-Long Course in Euclidean Geometry Must Go." Mathematics Teacher 100, no. 3 (October 2006): 165–68. http://dx.doi.org/10.5951/mt.100.3.0165.

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It is assumed that the geometry course refers to one that is commonly taught in the tenth school year. It is traditional Euclidean synthetic geometry, of 2- and 3-space, modified by an introduction of ruler and protractor axioms into the usual synthetic axioms. A unit of coordinate geometry of the plane is usually appended. It is a course that is reflected in textbooks prepared by the School Mathematics Study Group and in most commercial textbooks.

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20

Cabbolet, Marcoen J. T. F. "A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory." Axioms 10, no. 2 (June 14, 2021): 119. http://dx.doi.org/10.3390/axioms10020119.

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It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

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21

Bergh, Petter Andreas, and Marius Thaule. "The axioms forn–angulated categories." Algebraic & Geometric Topology 13, no. 4 (July 2, 2013): 2405–28. http://dx.doi.org/10.2140/agt.2013.13.2405.

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22

Struever, Nancy S. "Vico's Axioms: The Geometry of the Human World." Rhetorica 17, no. 2 (1999): 222–27. http://dx.doi.org/10.1525/rh.1999.17.2.222.

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23

Costa, Gustavo, and James Robert Goetsch. "Vico's Axioms: The Geometry of the Human World." Italica 74, no. 3 (1997): 427. http://dx.doi.org/10.2307/479952.

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24

RANI, RACHNA, RAKESH KUMAR, and R. K. NAGAICH. "Axioms of spheres in lightlike geometry of submanifolds." Proceedings - Mathematical Sciences 126, no. 4 (October 3, 2016): 613–21. http://dx.doi.org/10.1007/s12044-016-0300-9.

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25

Han, Sang-Eon. "Low-level separation axioms from the viewpoint of computational topology." Filomat 33, no. 7 (2019): 1889–901. http://dx.doi.org/10.2298/fil1907889h.

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The present paper studies certain low-level separation axioms of a topological space, denoted by A(X), induced by a geometric AC-complex X. After proving that whereas A(X) is an Alexandroff space satisfying the semi-T1 2 -separation axiom, we observe that it does neither satisfy the pre T1 2 -separation axiom nor is a Hausdorff space. These are main motivations of the present work. Although not every A(X) is a semi-T1 space, after proceeding with an edge to edge tiling (or a face to face crystallization) of Rn, n ? N, denoted by T(Rn) as an AC complex, we prove that A(T(Rn)) is a semi-T1 space. Furthermore, we prove that A(En), induced by an nD Cartesian AC complex Cn = (En,N,dim), is also a semi-T1 space, n ? N. The paper deals with AC-complexes with the locally finite (LF-, for brevity) property, which can be used in the fields of pure and applied mathematics as well as digital topology, computational topology, and digital geometry.

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Katz, Nets Hawk, and Keith M. Rogers. "On the polynomial Wolff axioms." Geometric and Functional Analysis 28, no. 6 (September 14, 2018): 1706–16. http://dx.doi.org/10.1007/s00039-018-0466-7.

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27

Bonanzinga, Maddalena, Dimitrina Stavrova, and Petra Staynova. "Combinatorial separation axioms and cardinal invariants." Topology and its Applications 201 (March 2016): 441–51. http://dx.doi.org/10.1016/j.topol.2015.12.053.

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28

Boekholt, Sven, and Markus Stroppel. "Independence of axioms for fourgonal families." Journal of Geometry 72, no. 1-2 (December 1, 2001): 37–46. http://dx.doi.org/10.1007/s00022-001-8568-y.

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29

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, and Klee Irwin. "Group Geometrical Axioms for Magic States of Quantum Computing." Mathematics 7, no. 10 (October 11, 2019): 948. http://dx.doi.org/10.3390/math7100948.

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Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).

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Lowen, R., and M. Sioen. "A note on separation in AP." Applied General Topology 4, no. 2 (October 1, 2003): 475. http://dx.doi.org/10.4995/agt.2003.2046.

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<p>It is our aim in this note to take a closer look at some separation axioms in the construct AP of approach spaces and contractions. Whereas lower separation axioms seem to be qualitative, the higher ones seem to have a quantitative nature. Also some characterizations for the corresponding epireectors will be given.</p>

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31

WYNN, KAREN, and PAUL BLOOM. "The Origins of Psychological Axioms of Arithmetic and Geometry." Mind & Language 7, no. 4 (December 1992): 409–20. http://dx.doi.org/10.1111/j.1468-0017.1992.tb00313.x.

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32

Lin, Shu-Kun. "Tenth Volume of Axioms and Why Axioms Was Launched." Axioms 10, no. 3 (June 23, 2021): 129. http://dx.doi.org/10.3390/axioms10030129.

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Pfeifer, Christian. "Finsler spacetime geometry in physics." International Journal of Geometric Methods in Modern Physics 16, supp02 (November 2019): 1941004. http://dx.doi.org/10.1142/s0219887819410044.

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Finsler geometry naturally appears in the description of various physical systems. In this review, I divide the emergence of Finsler geometry in physics into three categories: dual description of dispersion relations, most general geometric clock and geometry being compatible with the relevant Ehlers–Pirani–Schild axioms. As Finsler geometry is a straightforward generalization of Riemannian geometry there are many attempts to use it as generalized geometry of spacetime in physics. However, this generalization is subtle due to the existence of non-trivial null directions. I review how a pseudo-Finsler spacetime geometry can be defined such that it provides a precise notion of causal curves, observers and their measurements as well as a gravitational field equation determining the Finslerian spacetime geometry dynamically. The construction of such Finsler spacetimes lays the foundation for comparing their predictions with observations, in astrophysics as well as in laboratory experiments.

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Cieliebak, Kai, and Alexandru Oancea. "Symplectic homology and the Eilenberg–Steenrod axioms." Algebraic & Geometric Topology 18, no. 4 (April 26, 2018): 1953–2130. http://dx.doi.org/10.2140/agt.2018.18.1953.

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Nyikos, Peter, and John E. Porter. "Hereditarily strongly cwH and other separation axioms." Topology and its Applications 156, no. 2 (December 2008): 151–64. http://dx.doi.org/10.1016/j.topol.2008.05.021.

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Hung, H. H. "Weak separation axioms and weak covering properties." Topology and its Applications 158, no. 15 (September 2011): 1997–2004. http://dx.doi.org/10.1016/j.topol.2011.06.042.

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Coghetto, Roland. "Klein-Beltrami Model. Part II." Formalized Mathematics 26, no. 1 (April 1, 2018): 33–48. http://dx.doi.org/10.2478/forma-2018-0004.

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Summary Tim Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) have shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [2, 3, 15, 4]. With the Mizar system [1], [10] we use some ideas are taken from Tim Makarios’ MSc thesis [12] for formalized some definitions (like the tangent) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as a further development of Tarski’s geometry in the formal setting [9].

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38

Pambuccian, Victor. "Prolegomena to any theory of proof simplicity." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (January 21, 2019): 20180035. http://dx.doi.org/10.1098/rsta.2018.0035.

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By looking at concrete examples from elementary geometry, we analyse the manner in which the simplicity of proofs could be defined. We first find that, when presented with two proofs coming from mutually incompatible sets of assumptions, the decision regarding which one is simplest can be made, if at all, only on the basis of reasoning outside of the formal aspects of the axiom systems involved. We then show that, if the axiom system is fixed, a measure of proof simplicity can be defined based on the number of uses of axioms deemed to be deep or valuable, and prove a number of new results regarding the need to use at least three times some axioms in the proof of others. One such major example is Pappus implies Desargues, which is shown to require three uses of Pappus. A similar situation is encountered with Veblen's proof that the outer form of the Pasch axiom implies the inner form thereof. The outer form needs to be used at least three times in any such proof. We also mention the likely conflicting requirements of directness of a proof and the length of a proof. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

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39

Hyder, David Jalal. "Physiological Optics and Physical Geometry." Science in Context 14, no. 3 (September 2001): 419–56. http://dx.doi.org/10.1017/s0269889701000151.

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ArgumentHermann von Helmholtz’s distinction between “pure intuitive” and “physical” geometry must be counted as the most influential of his many contributions to the philosophy of science. In a series of papers from the 1860s and 70s, Helmholtz argued against Kant’s claim that our knowledge of Euclidean geometry was an a priori condition for empirical knowledge. He claimed that geometrical propositions could be meaningful only if they were taken to concern the behaviors of physical bodies used in measurement, from which it followed that it was posterior to our acquaintance with this behavior. This paper argues that Helmholtz’s understanding of geometry was fundamentally shaped by his work in sense-physiology, above all on the continuum of colors. For in the course of that research, Helmholtz was forced to realize that the color-space had no inherent metrical structure. The latter was a product of axiomatic definitions of color-addition and the empirical results of such additions. Helmholtz’s development of these views is explained with detailed reference to the competing work of the mathematician Hermann Grassmann and that of the young James Clerk Maxwell. It is this separation between 1) essential properties of a continuum, 2) supplementary axioms concerning distance-measurement, and 3) the behaviors of the physical apparatus used to realize the axioms, which is definitive of Helmholtz’s arguments concerning geometry.

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Cui, Shawn, Michael Freedman, and Zhenghan Wang. "Complexity classes as mathematical axioms II." Quantum Topology 7, no. 1 (2016): 185–201. http://dx.doi.org/10.4171/qt/75.

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41

VanOyen, Larry, and Rich Wilders. "Delving Deeper: How Is Geometry in Tune with Bach?" Mathematics Teacher 101, no. 1 (August 2007): 62–68. http://dx.doi.org/10.5951/mt.101.1.0062.

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For a mathematician, the study or creation of mathematics is a wonderfully exhilarating exercise in highly sophisticated puzzle solving using a small set of rules, which are called axioms. Geometry provides a great opportunity for high school students to experience the axiomatic nature of mathematics.

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42

Igusa, Kiyoshi. "Axioms for higher torsion invariants of smooth bundles." Journal of Topology 1, no. 1 (October 25, 2007): 159–86. http://dx.doi.org/10.1112/jtopol/jtm011.

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Kreuzer, Alexander. "A system of axioms for projective Hjelmslev spaces." Journal of Geometry 40, no. 1-2 (April 1991): 125–47. http://dx.doi.org/10.1007/bf01225880.

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Janelidze, George, and Manuela Sobral. "Strict monadic topology I: First separation axioms and reflections." Topology and its Applications 273 (March 2020): 106963. http://dx.doi.org/10.1016/j.topol.2019.106963.

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Han, Sang-Eon. "Semi-separation axioms of the infinite Khalimsky topological sphere." Topology and its Applications 275 (April 2020): 107006. http://dx.doi.org/10.1016/j.topol.2019.107006.

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Whitehead, Ian. "Affine Weyl group multiple Dirichlet series: type." Compositio Mathematica 152, no. 12 (November 8, 2016): 2503–23. http://dx.doi.org/10.1112/s0010437x16007715.

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We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac–Moody root system $\widetilde{A}_{n}$, generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is over the rational function field $\mathbb{F}_{q}(t)$, and is based upon four natural axioms from algebraic geometry. We prove that the four axioms yield a unique series with meromorphic continuation to the largest possible domain and the desired infinite group of symmetries.

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Van Lindt, Dirk, and Leopold Verstraelen. "A survey on axioms of submanifolds in Riemannian and Kaehlerian geometry." Colloquium Mathematicum 54, no. 2 (1987): 193–213. http://dx.doi.org/10.4064/cm-54-2-193-213.

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Schellenberg, Bjorn. "A proposal for a variation on the axioms of classical geometry." International Journal of Mathematical Education in Science and Technology 41, no. 3 (April 15, 2010): 311–21. http://dx.doi.org/10.1080/00207390903398390.

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Coghetto, Roland. "Klein-Beltrami Model. Part I." Formalized Mathematics 26, no. 1 (April 1, 2018): 21–32. http://dx.doi.org/10.2478/forma-2018-0003.

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Summary Tim Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [3], [4], [14], [5]. With the Mizar system [2], [7] we use some ideas are taken from Tim Makarios’ MSc thesis [13] for the formalization of some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as further development of Tarski’s geometry in the formal setting [6]. Note that the model presented here, may also be called “Beltrami-Klein Model”, “Klein disk model”, and the “Cayley-Klein model” [1].

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Burn, R. P. "Non-Desarguesian planes and weak associativity." Mathematical Gazette 101, no. 552 (October 16, 2017): 458–64. http://dx.doi.org/10.1017/mag.2017.127.

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During the 19th century various criticisms of Euclid's geometry emerged and alternative axiom systems were constructed. That of David Hilbert ([1], 1899) paid particular attention to the independence of the axioms, and it is his insights which have shaped many of the further developments during the 20th century.We can, from his insights, define an affine plane as a set of points, with distinguished subsets called lines such thatAxiom 1: Given two distinct points, there is a unique line containing them both.Axiom 2: Given a line L and a point, p, not contained in L, there is a unique line containing p which does not intersect L.Axiom 3: There exist at least three points, not belonging to the same line.

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