Proving Four-Dimensional Space Algorithms of Orthogonal Four-Sphere Six-Plane and Four-Dimensional Verticality—Application of Pythagorean Theorem of Four Dimensional Volume (Formula 7)

2020 ◽  
Vol 10 (01) ◽  
pp. 23-29
Author(s):  
国伟 蔡
1993 ◽  
Vol 8 (4) ◽  
pp. 241-249
Author(s):  
Haresh Lalvani

Complex design and morphologic problems, consisting of multiple parameters, can be described, modelled and indexed using a meta-morphological technique presented here. All design and morphologic variables can be mapped in Euclidean n-dimensional space, where n is the number of variables. This meta-space contains all the possible solutions to the problem and each point in this solution n-space is a candidate solution. The desired solution, represented by a specific point in this space and coded (addressed) by its n-dimension Cartesian co-ordinates, is ranked by the hyper-distance of this point from the origin. The hyper-distance itself provides a design index (or hyper-index) of the solution and can be determined by the known hyper-Pythagorean theorem. The procedure is recursive and applies to complex design problems which are hierarchical and composed of problem-within-problem-within-problems. Here the parameters are composed of subparameters, and the solutions are correspondingly mapped in a recursive, fractal n-cube composed of sub-cubes composed of sub-sub-cubes. The total composite index is determined by the recursive application of the hyper-Pythagorean theorem and represents a quantification of the morphological complexity of the design. The model is independent of the design problem, and has attractive possibilities for application in computer-aided design environments. The application is shown with the hypothetical selection of a space frame from a number of alternatives.


1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.


Author(s):  
David A. Agard ◽  
Yasushi Hiraoka ◽  
John W. Sedat

In an effort to understand the complex relationship between structure and biological function within the nucleus, we have embarked on a program to examine the three-dimensional structure and organization of Drosophila melanogaster embryonic chromosomes. Our overall goal is to determine how DNA and proteins are organized into complex and highly dynamic structures (chromosomes) and how these chromosomes are arranged in three dimensional space within the cell nucleus. Futher, we hope to be able to correlate structual data with such fundamental biological properties as stage in the mitotic cell cycle, developmental state and transcription at specific gene loci.Towards this end, we have been developing methodologies for the three-dimensional analysis of non-crystalline biological specimens using optical and electron microscopy. We feel that the combination of these two complementary techniques allows an unprecedented look at the structural organization of cellular components ranging in size from 100A to 100 microns.


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