scholarly journals Continuous flattening of the 2-dimensional skeleton of a regular 24-cell

2021 ◽  
Vol 112 (1) ◽  
Author(s):  
Jin-ichi Itoh ◽  
Chie Nara
Keyword(s):  
Convex Polyhedra ◽  
Regular Polytopes ◽  
Higher Dimensional ◽  
Buckminster Fuller

AbstractBellow theorem says that any polyhedron with rigid faces cannot change its volume even if it is flexible. The problem on continuous flattenig of polyhedra with non-rigid faces proposed by Demaine et al. was solved for all convex polyhedra by using the notion of moving creases to change some of the faces. This problem was extended to a problem on continuous flattening of the 2-dimensional skeleton of higher dimensional polytopes. This problem was solved for all regular polytopes except three types, the 24-cell, the 120-cell, and the 600-cell. This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton, which is related to the Jitterbug by Buckminster Fuller.

Higher Dimensional Periodic Table Of Regular And Semi-Regular Polytopes

1996 ◽  
Vol 11 (1-2) ◽  
pp. 155-171 ◽  
Author(s):  
Haresh Lalvani
Keyword(s):  
Periodic Table ◽  
Hyperbolic Spaces ◽  
Structural Knowledge ◽  
Space Structures ◽  
Regular Space ◽  
Regular Polytopes ◽  
Regular Polytope ◽  
Architectural Form ◽  
Hypercubic Lattice ◽  
Higher Dimensional

This paper presents a higher-dimensional periodic table of regular and semi-regular n-dimensional polytopes. For regular n-dimensional polytopes, designated by their Schlafli symbol {p,q,r,…u,v,w}, the table is an (n-1)-dimensional hypercubic lattice in which each polytope occupies a different vertex of the lattice. The values of p,q,r,…u,v,w also establish the corresponding n-dimensional Cartesian co-ordinates (p,q,r,…u,v,w) of their respective positions in the hypercubic lattice. The table is exhaustive and includes all known regular polytopes in Euclidean, spherical and hyperbolic spaces, in addition to others candidate polytopes which do not appear in the literature. For n-dimensional semi-regular polytopes, each vertex of this hypercubic lattice branches into analogous n-dimensional cubes, where each n-cube encompasses a family with a distinct semi-regular polytope occupying each vertex of each n-cube. The semi-regular polytopes are obtained by varying the location of a vertex within the fundamental region of the polytope. Continuous transformations within each family are a natural fallout of this variable vertex location. Extensions of this method to less regular space structures and to derivation of architectural form are in progress and provide a way to develop an integrated index for space structures. Besides the economy in computational processing of space structures, integrated indices based on unified morphologies are essential for establishing a meta-structural knowledge base for architecture.

scholarly journals High-dimensional Private Quantum Channels and Regular Polytopes

2021 ◽  
Vol 31 (2) ◽  
pp. 189
Author(s):  
Junseo Lee ◽  
Kabgyun Jeong
Keyword(s):  
Quantum State ◽  
Quantum Channel ◽  
Three Dimensional ◽  
Quantum Channels ◽  
Regular Polytopes ◽  
Quantum Fourier Transform ◽  
Quantum Communications ◽  
Regular Polyhedra ◽  
Higher Dimensional ◽  
Quantum Analog

As the quantum analog of the classical one-time pad, the private quantum channel (PQC) plays a fundamental role in the construction of the maximally mixed state (from any input quantum state), which is very useful for studying secure quantum communications and quantum channel capacity problems. However, the undoubted existence of a relation between the geometric shape of regular polytopes and private quantum channels in the higher dimension has not yet been reported. Recently, it was shown that a one-to-one correspondence exists between single-qubit PQCs and three-dimensional regular polytopes (i.e., regular polyhedra). In this paper, we highlight these connections by exploiting two strategies known as a generalized Gell-Mann matrix and modified quantum Fourier transform. More precisely, we explore the explicit relationship between PQCs over a qutrit system (i.e., a three-level quantum state) and regular 4-polytopes. Finally, we attempt to devise a formula for such connections on higher dimensional cases.

scholarly journals Piecewise Linear Sheaves

2019 ◽  
Author(s):  
Masaki Kashiwara ◽  
Pierre Schapira
Keyword(s):  
Vector Space ◽  
Piecewise Linear ◽  
Building Blocks ◽  
Convex Polyhedra ◽  
Triangulated Category ◽  
Real Vector ◽  
Proper Cone ◽  
Finite Dimensional ◽  
Higher Dimensional

Abstract On a finite-dimensional real vector space, we give a microlocal characterization of (derived) piecewise linear sheaves (PL sheaves) and prove that the triangulated category of such sheaves is generated by sheaves associated with convex polyhedra. We then give a similar theorem for PL $\gamma $-sheaves, that is, PL sheaves associated with the $\gamma $-topology, for a closed convex polyhedral proper cone $\gamma $. Our motivation is that convex polyhedra may be considered as building blocks for higher dimensional barcodes.

Spectroscopy-Based Mapping with Scanning Microwave Impedance Microscopy

2018 ◽  
Author(s):  
Peter De Wolf ◽  
Zhuangqun Huang ◽  
Bede Pittenger
Keyword(s):  
Single Point ◽  
Electrical Characterization ◽  
Principal Component ◽  
High Sensitivity ◽  
Data Cube ◽  
Nanometer Scale ◽  
Learning Approaches ◽  
Data Set ◽  
3D Data ◽  
Higher Dimensional

Abstract Methods are available to measure conductivity, charge, surface potential, carrier density, piezo-electric and other electrical properties with nanometer scale resolution. One of these methods, scanning microwave impedance microscopy (sMIM), has gained interest due to its capability to measure the full impedance (capacitance and resistive part) with high sensitivity and high spatial resolution. This paper introduces a novel data-cube approach that combines sMIM imaging and sMIM point spectroscopy, producing an integrated and complete 3D data set. This approach replaces the subjective approach of guessing locations of interest (for single point spectroscopy) with a big data approach resulting in higher dimensional data that can be sliced along any axis or plane and is conducive to principal component analysis or other machine learning approaches to data reduction. The data-cube approach is also applicable to other AFM-based electrical characterization modes.

HIGHER DIMENSIONAL ROBERTSON WALKER UNIVERSES INTERACTING WITH BRANS-DICKE FIELD

2020 ◽  
Vol 9 (10) ◽  
pp. 8545-8557
Author(s):  
K. P. Singh ◽  
T. A. Singh ◽  
M. Daimary
Keyword(s):  
Higher Dimensional

Tools of Transformation: Appropriate Technology in U.S. Countercultural Literature

2012 ◽  
Vol 44 (2) ◽  
pp. 75-93
Author(s):  
Peter Mortensen
Keyword(s):  
Low Cost ◽  
Appropriate Technology ◽  
Small Scale ◽  
Environmental Transformation ◽  
Buckminster Fuller ◽  
1960S And 1970S ◽  
The 1960S ◽  
Environmentally Sound ◽  
User Friendly ◽  
Second Wave

This essay takes its cue from second-wave ecocriticism and from recent scholarly interest in the “appropriate technology” movement that evolved during the 1960s and 1970s in California and elsewhere. “Appropriate technology” (or AT) refers to a loosely-knit group of writers, engineers and designers active in the years around 1970, and more generally to the counterculture’s promotion, development and application of technologies that were small-scale, low-cost, user-friendly, human-empowering and environmentally sound. Focusing on two roughly contemporary but now largely forgotten American texts Sidney Goldfarb’s lyric poem “Solar-Heated-Rhombic-Dodecahedron” (1969) and Gurney Norman’s novel Divine Right’s Trip (1971)—I consider how “hip” literary writers contributed to eco-technological discourse and argue for the 1960s counterculture’s relevance to present-day ecological concerns. Goldfarb’s and Norman’s texts interest me because they conceptualize iconic 1960s technologies—especially the Buckminster Fuller-inspired geodesic dome and the Volkswagen van—not as inherently alienating machines but as tools of profound individual, social and environmental transformation. Synthesizing antimodernist back-to-nature desires with modernist enthusiasm for (certain kinds of) machinery, these texts adumbrate a humanity- and modernity-centered post-wilderness model of environmentalism that resonates with the dilemmas that we face in our increasingly resource-impoverished, rapidly warming and densely populated world.

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

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