Foundations of Three-Dimensional Euclidean Geometry

2020 ◽  
Author(s):  
Izu Vaisman
Keyword(s):  
Euclidean Geometry ◽  
Three Dimensional

Enhancing Plane Euclidean Geometry with Three-Dimensional Analogs

2008 ◽  
Vol 102 (5) ◽  
pp. 394-398
Author(s):  
Alfred S. Posamentier ◽  
L. Raphael Patton
Keyword(s):  
Euclidean Geometry ◽  
Three Dimensional ◽  
Educational Process ◽  
Human Beings ◽  
The World

Human beings are born with an ability to sense, perceive, and then remember, and as we grow and develop this ability matures through exercise and practice. Most of us can recall learning to see how circles and lines are tangent to one another, how midpoints and perpendiculars partition triangles, and how one might prove conjectures about these. Learning how to see literally, how to make images of the world, and how to perceive the world more completely and accurately is part of that educational process.

The shock of the odd

2015 ◽  
Vol 48 (2) ◽  
pp. 353-356 ◽  
Author(s):  
Boris Jardine
Keyword(s):  
Euclidean Geometry ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Fourth Dimension ◽  
Lively Discussion ◽  
Avant Garde ◽  
Non Euclidean Geometry ◽  
The World ◽  
The One ◽  
Formal Innovation

When Linda Dalrymple Henderson's The Fourth Dimension and Non-Euclidean Geometry in Modern Art first appeared in 1983 it generated a lively discussion, most conspicuously in the pages of the journal Leonardo. Here was a book that undermined two of the central tenets of modernist theory: first that developments in art and science were linked not by any real connections or strong form of shared endeavour but by the fact that both partook of the modern spirit or zeitgeist; second, and more specifically, that Einsteinian relativity and cubism were in some way analogous embodiments of that spirit. By relentlessly pursuing the fate of two nineteenth-century developments – the non-Euclidean geometries and higher dimensions of her title – Henderson clearly showed that many of the avant-garde artworks so admired by critics for their formal innovation were at once more literal and more bizarre than anyone had previously suspected. Some were attempts to expound the ‘geometrical occult’ or to engage in multidimensional communion, some projected the enhanced intellect of ‘four-dimensional man’ and others explored the lonely but profound reaches of hyperspace. As she puts it in the ‘Re-introduction’ to this new edition of The Fourth Dimension, ‘these works function as “windows” on an invisible meta-reality of higher dimensions and etherial energies' (p. 27), and, elsewhere, ‘belief in a fourth dimension encouraged artists to depart from visual reality and to reject completely the one-point perspective system that for centuries had portrayed the world as three-dimensional’ (p. 492).

scholarly journals Introduction into the Extra Geometry of the Three–Dimensional Space I

2020 ◽  
Vol 5 (5) ◽  
pp. 538-544 ◽  
Author(s):  
Istvan Szalay ◽  
B. Szalay
Keyword(s):  
Euclidean Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Real Numbers ◽  
Axiom Systems ◽  
Three Dimensional Space

Using the theory of exploded numbers by the axiom–systems of real numbers and euclidean geometry, we introduce a geometry in the three–dimensional space which is different from the euclidean-, Bolyai – Lobachevsky- and spherical geometries. In this part the concept of extra-line and extra parallelism are detailed.

scholarly journals Introduction into Extra Geometry of the Three-Dimensional Space II

2020 ◽  
Vol 5 (8) ◽  
pp. 904-914
Author(s):  
Istvan Szalay ◽  
Balazs Szalay
Keyword(s):  
Euclidean Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Essential Difference ◽  
Real Numbers ◽  
Euclidean Planes ◽  
Axiom Systems ◽  
Three Dimensional Space

Using the theory of exploded numbers by the axiom-systems of real numbers and Euclidean geometry, we introduce concept of extra - plane of the three-dimensional space. The extra - planes are visible subsets of super-planes which are exploded Euclidean planes. We investigate the main properties of extra-planes. We prove more similar properties of Euclidean planes and extra-planes, but with respect the parllelism there is an essential difference among them.

The Art of Teaching: Teaching Solid Geometry

1943 ◽  
Vol 36 (3) ◽  
pp. 126-128
Author(s):  
Nancy C. Wylie
Keyword(s):  
Mathematics Teacher ◽  
Euclidean Geometry ◽  
Three Dimensional ◽  
Solid Geometry ◽  
Art Of Teaching ◽  
Spatial Imagination

This Article is corroborating and supplementing the timely suggestions made by James V. Bernardo in the January 1940 issue of The Mathematics Teacher, on the teaching of solid geometry. I say, timely; first, because of its contribution to the meager body of material on the teaching of these books of Euclidean geometry; second, because most instructors are ready to begin a new semester and desire all additional light on methods of presenting the three-dimensional concepts. Any aids for perfecting technique that will enable the instructor to help the pupil in developing his spatial imagination will, very probably, be received with enthusiasm.

scholarly journals DIMENSÕES FÍSICAS, GEOMETRIA, PERSPECTIVA, FILOSOFIA E ARTE

Sapere Aude ◽  
2019 ◽  
Vol 10 (19) ◽  
pp. 184-202
Author(s):  
Raquel Anna Sapunaru
Keyword(s):  
Sixteenth Century ◽  
Fifteenth Century ◽  
Euclidean Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Arts ◽  
Filippo Brunelleschi ◽  
The Many ◽  
Dimensionality Of Space ◽  
Three Dimensional Space

A ideia de um espaço tridimensional começou a se formar no século XV. Antes disso, em um mundo dominado pelo aristotelismo, o espaço era vinculado à superfície e não ao volume. Foi através das artes que essa realidade começou a mudar. A perspectiva racional, definida aqui como um recurso gráfico que utiliza o efeito visual de linhas convergentes para criar a ilusão de tridimensionalidade do espaço e das formas representadas sobre uma superfície plana de um papel ou tela, nascida a partir de uma retomada da geometria euclidiana, entrou em cena para ficar no século XVI. Entre os muitos nomes que poderiam ser citados, destacaram-se o matemático e filósofo John Dee, o arquiteto e designer Filippo Brunelleschi, e o pintor e matemático Pierro della Francesca. Através da combinação das ideias e realizações desses três atores é possível entender uma época de transição entre o antigo e o moderno, em termos de ciência e arte.PALAVRAS-CHAVE: Espaço tridimencional. Geometria. Perspectiva racional. Filosofia e arte. ABSTRACTThe idea of a three-dimensional space began to form in the fifteenth century. Before that, in a world dominated by Aristotelianism, space was bound to the surface and not to the volume. It was through the arts that this reality began to change. The rational perspective, defined here as a graphic resource that uses the visual effect of converging lines to create the illusion of three-dimensionality of space and forms represented on a flat surface of a paper or canvas, born from a resumption of Euclidean geometry, came into the scene to stay in the sixteenth century. Among the many names that could be cited were the mathematician and philosopher John Dee, the architect and designer Filippo Brunelleschi, and the painter and mathematician Pierro della Francesca. By combining the ideas and achievements of these three actors it is possible to understand a time of transition between the old and the modern, in terms of science and art.KEYWORDS: Three-dimensional space. Geometry. Rational Perspective. Philosophy and Art.

scholarly journals Using direct and contructive methods for the existence of origami models with given boundary conditions

2020 ◽  
Vol 01 (01) ◽  
pp. 54-71
Author(s):  
R. GERETSCHLAGER ◽  
◽  
S.L. KEELING ◽  
Keyword(s):  
Boundary Conditions ◽  
Euclidean Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Closed Curve ◽  
Closed Curves ◽  
Unit Square ◽  
Total Length ◽  
Three Dimensional Space

Whenever a unit square is folded to create an origami model in three-dimensional space, the edge of the paper forms a closed curve in space with a total length equal to four units. In this paper, some of the restrictions applicable to such resulting closed curves are derived in the case of classic origami models, in which none of the sections of the folded paper is curved in any way. This allows us to restrict the methods applied to those of classic euclidean geometry. Noting that it is of interest to determine origami models whose edges coincide with a polyline fulfilling the required conditions, we then proceed to show some methods for reconstructing the origami model if the boundary is known. Finally, some concrete reconstructions are demonstrated.

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