scholarly journals Formal Geometry and Bordism Operations

2018 ◽  
Author(s):  
Eric Peterson
Keyword(s):  
Formal Geometry

Chess in the Geometry Classroom

1975 ◽  
Vol 68 (1) ◽  
pp. 71-72
Author(s):  
Nancy C. Whitman
Keyword(s):  
Physical Environment ◽  
Euclidean Geometry ◽  
Deductive System ◽  
Advance Organizer ◽  
Current Interest ◽  
Effective Technique ◽  
Formal Geometry ◽  
Chess Game ◽  
Chess Playing

The current interest in chess playing in this country prompts me to share a very effective technique I have used in introducing the study of formal geometry. Basically, it uses the chess game as an “advance organizer” of Euclidean geometry viewed as a deductive system. Of course, this is but one of several possible views of Euclidean geometry. For example, another view is that geometry is an abstraction of man's physical environment.

scholarly journals RATIONALITY PROPERTIES OF MANIFOLDS CONTAINING QUASI-LINES

2003 ◽  
Vol 14 (10) ◽  
pp. 1053-1080 ◽  
Author(s):  
PALTIN IONESCU ◽  
DANIEL NAIE
Keyword(s):  
Open Subset ◽  
Basic Question ◽  
Projective Manifold ◽  
Rational Varieties ◽  
Direct Sum ◽  
Formal Geometry ◽  
Rationally Connected ◽  
Birational Invariant ◽  
Small Deformations

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification [Formula: see text] that contains a quasi-line, i.e. a smooth rational curve whose normal bundle is a direct sum of copies of [Formula: see text]. For manifolds containing quasi-lines, a sufficient condition of rationality is exploited: there is a unique quasi-line from a given family passing through two general points. We define a numerical birational invariant, e(X), and prove that X is rational if and only if e(X)=1. If X is rational, there is a modification [Formula: see text] which is strongly-rational, i.e. contains an open subset isomorphic to an open subset of the projective space whose complement is at least 2-codimensional. We prove that strongly-rational varieties are stable under smooth, small deformations. The argument is based on a convenient characterization of these varieties. Finally, we relate the previous results and formal geometry. This relies on ẽ(X, Y), a numerical invariant of a given quasi-line Y that depends only on the formal completion [Formula: see text]. As applications we show various instances in which X is determined by [Formula: see text]. We also formulate a basic question about the birational invariance of ẽ(X, Y).

scholarly journals On Right and Wrong Drawings

2016 ◽  
Vol 4 (1-2) ◽  
pp. 1-38 ◽  
Author(s):  
Jan Koenderink ◽  
Andrea van Doorn ◽  
Baingio Pinna ◽  
Robert Pepperell
Keyword(s):  
Visual Arts ◽  
Vantage Point ◽  
Linear Perspective ◽  
Minor Effect ◽  
Fixed Position ◽  
Making Sense ◽  
Formal Geometry ◽  
Art Works ◽  
A Minor ◽  
Pictorial Experience

Are pictorial renderings that deviate from linear perspective necessarily ‘wrong’? Are those in perfect linear perspective necessarily ‘right’? Are wrong depictions in some sense ‘impossible’? Linear perspective is the art of the peep show, making sense only from one fixed position, whereas typical art works are constructed and used more like panel presentations, that leave the vantage point free. In the latter case the viewpoint is free; moreover, a change of viewpoint has only a minor effect on pictorial experience. This phenomenologically important difference can be made explicit and formal, by considering the effects of panning eye movements when perusing scenes, and of changes of viewpoint induced by translations with respect to pictorial surfaces. We present examples from formal geometry, photography, and the visual arts.

Geometry of Pictorial Relief

2018 ◽  
Vol 4 (1) ◽  
pp. 451-474 ◽  
Author(s):  
Jan J. Koenderink ◽  
Andrea J. van Doorn ◽  
Johan Wagemans
Keyword(s):  
Conceptual Understanding ◽  
Visual Awareness ◽  
Color Pattern ◽  
Formal Models ◽  
Geometrical Properties ◽  
Formal Geometry ◽  
Planar Substrate ◽  
Depth Dimension ◽  
Spatial Entity

Pictorial relief is a quality of visual awareness that happens when one looks into (as opposed to at) a picture. It has no physical counterpart of a geometrical nature. It takes account of cues, mentally identified in the tonal gradients of the physical picture—pigments distributed over a planar substrate. Among generally recognized qualities of relief are color, pattern, texture, shape, and depth. This review focuses on geometrical properties, the spatial variation of depth. To be aware of an extended quality like relief implies a “depth” dimension, a nonphysical spatial entity that may smoothly vary in a surface-like manner. The conceptual understanding is in terms of formal geometry. The review centers on pertinent facts and formal models. The facts are necessarily so-called brute facts (i.e., they cannot be explained scientifically). This review is a foray into the speculative and experimental phenomenology of the visual field.

The Teaching of Mathematics in the Elementary and the Secondary School

1909 ◽  
Vol 1 (4) ◽  
pp. 121-122
Author(s):  
Arthur Sullivan Gale
Keyword(s):  
Secondary School ◽  
Experimental Work ◽  
Euclidean Geometry ◽  
Geometric Analysis ◽  
Algebraic Equations ◽  
Algebraic Expressions ◽  
Formal Geometry ◽  
Non Euclidean Geometry ◽  
Teaching Of Mathematics ◽  
Class Room

The chapter on geometry (pp. 257-291) is excellent. lt presents ideas on geometric analysis, concrete and formal geometry, methods for treating problems, modern geometry, and non-Euclidean geometry. Especially important is the discussion of problems which lead to algebraic equations and the construction of simple algebraic expressions. A timely plea is made for experimental work and the usc of models and apparatus. As an example of their value, a Rochester teacher exhibited a sextant before a class one morning. A pupil borrowed it for the noon hour and became so enthusiastic in its use that he “cut” his afternoon classes to do some rough surveying. Contrast the interest which the inc;trument developed with the lack of enthusiasm which causes so many absences from the mathematical class-room! The chapter closes with an analysis of trigonometry and suggestions as to where its various parts should be taught.

Should Formal Geometry be Taught in the Elementary Schools? If so, to What Extent?

1912 ◽  
Vol 4 (4) ◽  
pp. 144-149
Author(s):  
D. J. Kelly
Keyword(s):  
Elementary Schools ◽  
Mathematics Teacher ◽  
Formal Geometry ◽  
Do So

In appearing before this assembly I feel somewhat like an impostor, for I am not a mathematics teacher nor have I ever been one. Neither do I make any claim as a mathematician but am merely a plain superintendent of schools, somewhat young in experience and a trifle old-fashioned in ideas. As such I speak this afternoon and should you disagree with anything that is said you are at liberty to do so for “my hat is not in the ring” nor have I any fears of “recall.”

Sign in / Sign up

Export Citation Format

Share Document