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Transformational Geometry
Section 5.5 presented background information for Pieri’s use of transformational methods in his pioneering axiomatization of projective geometry,...
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Learning orientations: a discrete geometry model
In the mammalian brain, many neuronal ensembles are involved in representing spatial structure of the environment. In particular, there exist cells...
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Many-Valued Logical Hexagons in a 3-Oppositional Trisimplex
The present paper’s aim is to construct a “3-oppositional trisimplex” (i.e., the tri-simplicial counterpart of the “logical hexagon” (1950), the most...
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Logical Oppositions in Avicenna’s Hypothetical Logic
In his hypothetical logic, Avicenna introduces new kinds of hypothetical propositions by using quantifications ranging over situations and...
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Stitching Data: Recovering a Manifold’s Geometry from Geodesic Intersections
Let ( M , g ) be a Riemannian manifold with boundary. We show that knowledge of the length of each geodesic, and where pairwise intersections occur...
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Pieri and Projective Geometry
Projective geometry can be described as the geometry of the straightedge, in comparison with Euclid’s geometry of straightedge and compass.
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Visual Representations of Euclidean Geometry: Diagrammatic Reasoning in Oliver Byrne’s Work
Around the half of the nineteenth century the Irish mathematician and civil engineer Oliver Byrne published a peculiar version of Euclid’s Elements...
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From Euler Diagrams in Schopenhauer to Aristotelian Diagrams in Logical Geometry
In this paper I explore the connection between Schopenhauer’s Euler diagrams and the Aristotelian diagrams that are studied in contemporary logical...
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Logic, Spatial Algorithms and Visual Reasoning
Spatial and diagrammatic reasoning is a significant part not only of logical abilities, but also of logical studies. The authors of this paper...
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Disjunctive Bases of Applied Algebras of Sets and Their Use in Problems of Combinatorial Geometry
In this article, the concept of applied algebra of sets and a standard disjunctive basis of an applied algebra of sets are introduced. The algorithm...
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Computer Algebra Systems and Dynamic Geometry for Mathematical Thinking
For more than three decades, the use of Computer Algebra Systems (CAS) and Dynamic Geometry Environments (DGE) has introduced new possibilities in...
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Algorithmic Modelling as a Key Tool for Ribbed Vault Geometry
The paper proposes the first results of an ongoing research on the geometry of ribbed cross vaults starting from the principles and theories...
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Algebraic Geometry over Algebraic Structures. VIII. Geometric Equivalences and Special Classes of Algebraic Structures
This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven...
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Pre-College Geometry, Mensuration, Trigonometry, Surveying, and Navigation 1607–1865
This chapter analyzes pre-college education developments in geometry, mensuration, trigonometry, surveying, and navigation between 1607 and 1865, in...
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Fractal Geometry in Architecture
Fractal geometry is a product of fractal theory, a mathematical approach that describes the way space is filled by figures or objects. A fractal...
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1900s–1930s: Branch Curves and the Italian School of Algebraic Geometry
In contrast to the separate beginnings during the nineteenth century presented in the previous chapter, the twentieth century saw the consolidation...
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Geometry and Mathematics in Timurid Architecture: Abu’l-Wafa and Shirazi
The treatises of Mohammad Abu’l-Wafa al-Buzjani (tenth c.) have influenced many architects, who utilized his innovative geometric principles and...
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Cayley–Klein geometries and projective-metric geometry
Cayley and Klein discovered in the 19th century that Euclidean and non-Euclidean geometries can be introduced as geometries living inside of a...
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Two Paths to Logical Consequence: Pieri and the Peano School
This chapter1 has two main goals. First, it will explore the “negative” avenue leading from the concepts of independence and consistency to that of...