23 papers · ranked by Valyu relevance
James T. Todd, Alexander A. Petrov
Shape is an interesting property of objects because it is used in ordinary discourse in ways that seem to have little connection to how it is typically defined in mathematics. The present article describes how the concept of shape can be grounded within Euclidean and non-Euclidean geometry and also to human perception.…
Patrick D. Barry, Anthony G. O’Farrell
At present, the NCCA presents the mathematics curriculum in terms of strands. For primary level the five strands are labelled Number, Algebra, Shape and Space, Measures, and Data. For secondary level they are (1) data, statistics and probability, (2) geometry and trigonometry, (3) number and measure, (4) algebra and…
John G. Ratcliffe, Sophia Stone, Steven T. Tschantz
In this paper, we analyze the two geometrical passages in Plato's Meno, (81c – 85c) and (86e4 – 87b2), from the points of view of a geometer in Plato's time and today. We give, in our opinion, a complete explanation of the difficult second geometrical passage. Our explanation solves an ingenious geometry puzzle that…
Vincenzo De Risi
The paper lists several editions of Euclid’s Elements in the Early Modern Age, giving for each of them the axioms and postulates employed to ground elementary mathematics.
Peter M. Johnson
The initial techniques developed in Euclid's Elements, well before the use of the parallel postulate, are reexamined in order to clarify even the most obscure details, particularly those related to equality, superposition and angle comparison. Some commentary on modern developments is included. The known but often…
Taiping Zeng, Ming Bo Cai
The representation of geometric structures of one’s surroundings is key to self-localization during human spatial navigation. However, the spatial organization of geometry representation in the visual system has not been fully characterized. By modeling the synchronized brain activity from participants watching…
Eldar Straume
This is an expository treatise on the development of the classical geometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of…
Eduardo N. Giovannini, Abel Lassalle‐Casanave
A crucial trend of nineteenth-century mathematics was the search for pure foundations of specific mathematical domains by avoiding the obscure concept of magnitude. In this paper, we examine this trend by considering the “fundamental theorem” of the theory of plane area: “If a polygon is decomposed into polygonal parts…
Akihito Maruya, Qasim Zaidi
Judging poses, sizes and shapes of objects accurately is necessary for organisms and machines to operate successfully in the world. Retinal images of 3D objects are mapped by the rules of projective geometry, and preserve the invariants of that geometry. Since Plato, it has been debated whether geometry is innate to…
René De Vogelaere
The author of this monograph was my father, Professor Ren´e De Vogelaere. He received his PhD in Mathematics in 1948 from the University Louvain, Belgium. Shortly after graduation, he immigrated to Canada and taught at l'Universit´e Laval in Quebec, followed by Notre Dame in South Bend, Indiana and then the University…
Semir Zeki, Zachary F Hale, Ahmad Beyh, Samuel E Rasche
There are different definitions of axioms, but the one that seems to have general approval is that axioms are statements whose truths are universally accepted but cannot be proven; they are the foundation from which further propositional truths are derived. Previous attempts, led by David Hilbert, to show that all of…
Günther Eder
In recent years, several scholars have been investigating Frege’s mathematical background, especially in geometry, in order to put his general views on mathematics and logic into proper perspective. In this article I want to continue this line of research and study Frege’s views on geometry in their own right by…
Zoltán Kovács
GeoGebra [8] is a well known dynamic geometry software package with millions of users worldwide. One of its main purposes is to visualize geometric invariants. Recently GeoGebra has been supporting investigation of geometric constructions also symbolically by exploiting the strength of the embedded computer algebra…
Stefan Vuckovic
Nearly all electronic structure simulations begin with obtaining approximate geometries, making a systematic quantification of errors in approximate molecular structures of key importance. Recently, the geometric energy offset (GEO) framework based on a single and natural measure for quantifying and analysing these…
Mahwish Kittur, Agnes Zhang, Nessa V. Bryce, Sami R. Yousif
Human spatial representations are often assumed to represent Euclidean properties such as length, distance, and angle. Here we test an alternative (but not mutually exclusive) possibility – that spatial memory is structured primarily around topological relations. Across four experiments, adults and children memorized…
Yuval Hart, L. Mahadevan
The perception of the noisy visual world around us naturally combines geometry and probability with psychophysics. So how do we perceive geometric objects from a probabilistic perspective, i.e. infer randomness in a spatial setting ? To test this psychophysically, we use a set of simple experiments to distinguish…
Alexander Bucksch, Acheampong Atta-Boateng, Akomian Fortuné Azihou, Mathilde Balduzzi + 34 more
Plant morphology is inherently mathematical in that morphology describes plant form and architecture with geometrical and topological descriptors. The geometries and topologies of leaves, flowers, roots, shoots and their spatial arrangements have fascinated plant biologists and mathematicians alike. Beyond providing…
Authors not listed
Previously we posited that a systematic and general description of stereoisomerism could be based upon the principles of the polytopal rearrangement model. The most daunting challenge to this end is to comprehensively describe all possible geometries for arbitrary n-coordinate centres, ABn, and for this we have…
Authors not listed
The diverse structures of metal-organic frameworks (MOFs) originate from the coordination chemistry of ionic metal-ligand bonds, while covalent organic frameworks (COFs) leverage covalent bonding. Between these extremes, extended structures based on metalloids are comparatively rare. This paucity obscures how the…
Maria Chiara di Gregorio, Vivek Singh, Linda J. W. Shimon, Michal Lahav + 1 more
The symmetry of a crystal’s morphology usually reflects the symmetry of the crystallographic packing. For single-crystals, the space and point groups allow only a limited number of mathematical descriptions of the morphology (forms), all of which are convex polyhedrons. In contrast, concave polyhedrons are a hallmark…
Authors not listed
The range of geometric configurations possible for an arbitrary coordination centre ABn, is a key aspect of stereoisomerism and molecular geometry in general. Despite this, much focus within Chemistry concerning ABn geometries has not taken a rigorous and holistic approach. Using a precise definition for configurations…
Jixin Chen
I have been teaching physical chemistry to undergraduate students for several years now. I have observed that my interpretation of vector-matrix calculations and complex calculations has not been well received by the chemistry students. The traditional vector notation of adding an arrow on top of a variable and the…
Authors not listed
Nonlinear monotonically increasing bounded functions help to visualize and analyze data on various scales. However, many monotonic functions such as logarithm or power laws have either function values or derivatives that become unbounded at some regions of the $x-$ axis. On the other hand, sigmoid or hyperbolic…