How to see a cube moving into its mirror image

2012 ◽  
Vol 96 (537) ◽  
pp. 471-479
Author(s):  
Ester Dalvit ◽  
Domenico Luminati
Keyword(s):  
Euclidean Space ◽  
Dimensional Space ◽  
Rigid Motion ◽  
Mirror Image ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Other Hand ◽  
Rigid Motions

In n-dimensional Euclidean space no reflection with respect to a hyperplane can be realised by a rigid motion. But this is possible if we allow rigid motions in (n + 1)-dimensional space. These notes show a way to visualise a rigid motion of a cube in 4-dimensional space that flips the cube ‘as the page of a book’.The two terms rigid motion and isometry are sometimes used as synonyms. Yet they do refer to different concepts. The first one has a purely kinematic connotation: the swing of a door or the movement of a piece of furniture pushed over the floor are described by rigid motions. On the other hand to ensure that two figures are isometric it is enough that there exists a correspondence between their points that maintains the relative distances.

Development of the Perception of Invariants: Substance and Shape

Perception ◽  
1979 ◽  
Vol 8 (6) ◽  
pp. 609-619 ◽  
Author(s):  
Eleanor J Gibson ◽  
Cynthia J Owsley ◽  
Arlene Walker ◽  
Jane Megaw-Nyce
Keyword(s):  
Shape Change ◽  
Rigid Motion ◽  
The Other ◽  
Early Age ◽  
Event Sequence ◽  
Invariant Property ◽  
Other Hand ◽  
Rigid Motions ◽  
Invariant Properties

Three experiments investigated the perception of substance and shape as invariant properties of objects by three-month-old infants. In experiment 1, infants were habituated to two differently shaped objects undergoing a rigid motion. After habituation of the infants, the objects were presented undergoing a different rigid motion, or undergoing a deforming motion, or undergoing the same rigid motion. Habituation was maintained to the new rigid motion, indicating that the two rigid motions were perceived as sharing an invariant property. Dishabituation, on the other hand, occurred when a deforming motion followed a rigid one. In experiment 2, infants were habituated to one shape undergoing two different rigid motions. After habituation, the shape was changed but the same two motions continued. Dishabituation occurred, compared to a group with no shape change, indicating that shape is distinguished as an invariant property over two rigid motions. In experiment 3, habituation to a shape undergoing two rigid motions was followed by a new shape presented motionless, or the same shape presented motionless. Cessation of motion did not prevent recognition of shape as invariant. Two properties of an object, substance and shape, thus appear to be detectable as invariant in an event sequence, an instance of ‘phenomenal doubling’ at an early age.

Relationship between Frames of Reference and Mirror-Image Reversals

Perception ◽  
10.1068/p5529 ◽  
2007 ◽  
Vol 36 (7) ◽  
pp. 1049-1056 ◽  
Author(s):  
Hirokazu Yoshimura ◽  
Tatsuo Tabata
Keyword(s):  
Mirror Image ◽  
Frame Of Reference ◽  
The Other ◽  
Frames Of Reference ◽  
Physical Aspect ◽  
Other Hand ◽  
Extended Problem ◽  
Left Right Asymmetry ◽  
Common Frame Of Reference ◽  
Intrinsic Frame

The mirror puzzle related to the perception of mirror images as left–right reversed can be more fully understood by considering an extended problem that includes also the perception of mirror images that are not left–right reversed. The purpose of the present study is to clarify the physical aspect of this extended problem logically and parsimoniously. Separate use of the intrinsic frame of reference that belongs to the object and one that belongs to its mirror image always leads to the perception of left–right reversal when the object has left–right asymmetry; on the other hand, the perception of left–right nonreversal is always due to the application of a common frame of reference to the object and its mirror image.

LP -bound for the Fourier Transform of Surface-Carried Measures Supported on Hypersurfaces with D∞ Type Singularities

2020 ◽  
pp. 350-359
Author(s):  
Nigina A. Soleeva
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Convex Surfaces ◽  
The Fourier Transform

Estimate for Fourier transform of surface-carried measures supported on non-convex surfaces of three-dimensional Euclidean space is considered in this paper.The exact convergence exponent was found wherein the Fourier transform of measures is integrable in tree-dimensional space. This result gives an answer to the question posed by Erd¨osh and Salmhofer

Equivalence of Cables Of Mutants of Knots

1989 ◽  
Vol 41 (2) ◽  
pp. 250-273 ◽  
Author(s):  
Józef H. Przytycki
Keyword(s):  
Mirror Image ◽  
Alexander Polynomial ◽  
The Other ◽  
Homfly Polynomial ◽  
Similar Formula ◽  
Conway Polynomial ◽  
Other Hand ◽  
Limited Possibility

There is the nice formula which links the Alexander polynomial of (m, k)-cable of a link with the Alexander polynomial of the link [5] [36] [38]. H. Morton and H. Short investigated whether a similar formula holds for the Jones-Conway (Homfly) polynomial and they found that it is very unlikely. Morton and Short made many calculations of the Jones-Conway polynomial of (2, q)-cables along knots (2 was chosen because of limited possibility of computers) and they get very interesting experimental material [24], [25]. In particular they found that using their method they were able to distinguish some Birman [4] and Lozano-Morton [22] examples (all which they tried) and the 942 knot (in the Rolfsen [37] notation) from its mirror image. On the other hand they were unable to distinguish the Conway knot and the Kinoshita-Terasaka knot.

scholarly journals The number of partitions of a set of N points in k dimensions induced by hyperplanes

1967 ◽  
Vol 15 (4) ◽  
pp. 285-289 ◽  
Author(s):  
E. F. Harding
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
General Position ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Unordered Pair

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

Crystallography, geometry and physics in higher dimensions. V. Polar and mono-incommensurate point groups in the four-dimensional space %E4

1989 ◽  
Vol 45 (2) ◽  
pp. 187-193 ◽  
Author(s):  
R. Veysseyre ◽  
D. Weigel
Keyword(s):  
Euclidean Space ◽  
Dimensional Space ◽  
Physical Space ◽  
Higher Dimensions ◽  
Dimensional Euclidean Space ◽  
Acta Cryst ◽  
Magnetic Structures ◽  
Convenient Means ◽  
Incommensurate Structures ◽  
Point Groups

The crystallographic point groups of the four-dimensional Euclidean space {\bb E}4are a convenient means of studying some crystallized solids of physical space, for instance the groups of magnetic structures and the groups of mono-incommensurate structures, as is demonstrated by a simple example. The concept of polar crystallographic point groups defined here in {\bb E}4, and also in {\bb E}nenables the list and the WPV notation {geometric symbol of Weigel, Phan & Veysseyre [Acta Cryst.(1987), A43, 294-304]} of these special structures to be stated in a more precise way. This paper is especially concerned with the mono-incommensurate structures while a discussion on magnetic structures will be published later.

scholarly journals The Third Dimention in a Landscape

Secreta Artis ◽  
2021 ◽  
pp. 74-82
Author(s):  
Daria Vladimirovna Fomicheva
Keyword(s):  
Land Surface ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Landscape Painting ◽  
The Third ◽  
Soviet Art ◽  
Other Hand ◽  
Third Dimension ◽  
Three Dimensional Space

The present study examines the principles of conveying the third dimension in landscape painting. The author analyzes the recommendations provided in J. Littlejohns’ manual entitled “The Composition of a Landscape” [London, 1931]. J. Littlejohns describes four methods of showing depth in a landscape painting, each illustrated with pictorial composition schemes: 1) portrayal of long roads, which allows one to unveil the plasticity of the land surface; 2) creation of a “route” for the viewer by means of a well-thought-out arrangement of natural landforms; 3) introduction of vertically and horizontally flowing streams of water on different picture planes; 4) depiction of cloud shadows on a distinctly hilly landscape. The author of the article compares the schemes contained in the manual of J. Littlejohns with the works of G. G. Nissky, which enables readers to comprehend and reflect on the compositions of the masterpieces created by a prominent figure in Soviet art; on the other hand, Nissky’s landscape paintings open for a deeper understanding of the meaning and effectiveness of the methods proposed by J. Littlejohns. The outlined composition techniques are certainly relevant for contemporary artists (painters, graphic artists, animators, designers, etc.) as they make it possible to achieve the plastic expressiveness of a three-dimensional space in a twodimensional image.

scholarly journals On the existence of four or more curved foldings with common creases and crease patterns

2021 ◽  
Author(s):  
A. Honda ◽  
K. Naokawa ◽  
K. Saji ◽  
M. Umehara ◽  
K. Yamada
Keyword(s):  
Euclidean Space ◽  
Closed Curve ◽  
The Other ◽  
Singular Set ◽  
Other Hand ◽  
Paper Folding ◽  
Simple Arc ◽  
Congruence Classes

AbstractConsider an oriented curve $$\Gamma $$ Γ in a domain D in the plane $${\varvec{R}}^2$$ R 2 . Thinking of D as a piece of paper, one can make a curved folding in the Euclidean space $${\varvec{R}}^3$$ R 3 . This can be expressed as the image of an “origami map” $$\Phi :D\rightarrow {\varvec{R}}^3$$ Φ : D → R 3 such that $$\Gamma $$ Γ is the singular set of $$\Phi $$ Φ , the word “origami” coming from the Japanese term for paper folding. We call the singular set image $$C:=\Phi (\Gamma )$$ C : = Φ ( Γ ) the crease of $$\Phi $$ Φ and the singular set $$\Gamma $$ Γ the crease pattern of $$\Phi $$ Φ . We are interested in the number of origami maps whose creases and crease patterns are C and $$\Gamma $$ Γ , respectively. Two such possibilities have been known. In the authors’ previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possible values for the number N of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if C is a non-closed simple arc, then $$N=4$$ N = 4 if and only if both $$\Gamma $$ Γ and C do not admit any symmetries. On the other hand, when C is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.

Reading The Time of Trimming under the Desk of Religious Zionism: Haim Beʾer and National-Religious Identity

Zutot ◽  
2014 ◽  
Vol 11 (1) ◽  
pp. 6-17
Author(s):  
Yael Shenker
Keyword(s):  
National Identity ◽  
Religious Identity ◽  
Mirror Image ◽  
Religious Discourse ◽  
Religious Community ◽  
National Identities ◽  
The Other ◽  
Other Hand ◽  
The One ◽  
The Relationship

This article addresses Israeli novelist Haim Beʾer’s relation to national-religious identity and the rifts and the pain it causes him, as can be discerned from his fiction and journalism, and certainly from interviews with him. His relation to national-religious identity also reflects a sort of mirror image, at times inverted, of the relationship between religious and national identities. Beʾer’s movement between religious community and nation criticizes on the one hand prevalent conceptions of secularization and national identity in Zionist discourse, and, on the other hand, conceptions of redemption in religious discourse.

scholarly journals Equation of the fixed membrane in then-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various norms

1979 ◽  
Vol 2 (2) ◽  
pp. 337-340
Author(s):  
Yves Biollay
Keyword(s):  
Dimensional Space ◽  
The Other ◽  
Other Hand ◽  
Fixed Membrane

We show in this paper that the sequence{max|uk|}, where theukare the eigenfunctions of the problemΔu+λu=0inD⊂Rnandu=0on∂D, is not bounded generally if one imposes the norm∫Du2p(x)dx=1,p=(1),2,3,…. The same holds with the norm∫D|gradu|2pdx=1whenn>4p−1. On the other hand, ifD⊂R2, resp.R3the norm∫D|gradu|2dx=1impliesmax|uk|→k→∞0, resp.max|uk|=0(1).

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