Conceptualizing Curse of Dimensionality with Parallel Coordinates

International Journal of …, 2009

The concept of dimension, one of the most fundamental ideas in mathematics, is firmly rooted in the basis of the school geometry in such a way that mathematics teachers rarely feel the need to mention anything about it. However, the concept of dimension is far from being fully understood by students, even at the college level. In this paper, we examine whether the Cartesian x-y plane is responsible for student difficulty in estimating the value of the dimension of an object, or is it only students misconceptions about dimension that lead them to a false estimation of the value of the dimension of various objects. A second question discussed in this paper examines whether the system of coordinates acts as an epistemological obstacle or whether it has only a didactical character.

2012

The Historical Background of Dimensionality as far as I know, the first modern mathematician conscious about dimensionality or space dimension is Bernhard Riemann. In his third doctoral paper(, Friedrich Gauss reviewed it and gave him the oral examination for it, then, he was so fascinated by the theory that he fell down a gutter on the way home after the interview.) he established so called in today Riemann Geometry. This is also known as mathematics Albert Einstein applied it when he established his theory of general relativity. In his thesis, he offered a tricky but profound idea of dimensionality by applying Pythagorean theorem; we know it is defined by a^2+b^2=c^2 in 2-dimensional space, and it is actually a^2+b^2+c^2=d^2 in 3-dimensional space. So he inductively generalized it as a^2+b^2+c^2+d^2+⋯=z^2 in higher dimensional space. Such a concept of higher dimensional space is succeeded to Einstein inspired by Theodor Kaluza and Oscar Klein in the process of establishing his unified theory including general relativity in his later life. It is also known as the Kaluza-Klein theory in today's theoretical physics. The K-K theory insists on that; there is higher dimensional space more than our 3-dimesnional space like 4, 5, 6, 7 dimensional one, but we can't detect it because it is compactified very very small. These days, physicists consider that there is wider space but we can't move to there just like an ant stuck to the wall(2-dimensional plane) don't know 3-dimensional space; in other words, 2-dimensional space is subset of 3-dimensional space. However, these ideas are really true? In a Dawn of October 5th in 1991 One night of October 4th in 1991, I was thinking of a strangeness why and how photon or electron have a duality of particle and wave in my bed; certainly quantum field theory answers the question by introducing various convenient fields, but I wanted to think of the duality without such fields or so called "mode". I couldn't fell asleep yet in the midnight because I was still pondering of its strangeness. Afterward, the time dawn breaks was coming, but still imagining a picture that the circle was growing without medium. At the time, God whispered it to me(; suddenly I got an inspiration.) That was a new idea of coordinate geometry based on quite a different concept from which Rene Descartes got about 400 years ago!

2018

Exploring the Meta-anomaly in Mathematics and its relation to Pascal's Triangle as Information Infrastructure for Levels of the Realms of Experience -- Abstract: Looking into Higher Dimensionality for its meaning and attempting to understand Celestial Causation in traditional sciences. -- Key Words: Mathematics, Systems Science, Meaning, Special Systems, Schemas, Meta-anomaly, Foundations, Pascal Triangle, Higher Dimensions, Celestial Causation, Traditional Sciences -- New version 2018.8.30

Here is analyzed the dimensionality of our Universe on the basis of multi space conception. The new space conception proposes that space is the manifestation of a force field. Space does not exist independently from a force field. Hence each forcefield has its own eigenspace with different dimensionality.

Oasis Journal, 2018

Short story published in Oasis Journal Vol. 3, May 2018.

2023

Dimensional analysis is a powerful tool for unraveling the fundamental relationships that govern the physical world. This paper takes a comprehensive journey through the concept of dimensions, their powers, and the intriguing transitions between them. These dimensions, including length, width, height, and time, are the building blocks of our physical reality. By exploring the shift between lower and higher dimensions, new attributes and powers emerge, enriching our understanding of the cosmos. This study delves into the mathematics behind quantum mechanics and quantum gravity, emphasizing the connections between energy, length scales, and fundamental constants. Notably, it elucidates the role of the fourth dimension, often associated with time. Through careful analysis, this paper demonstrates the prowess of dimensional analysis as a tool to investigate the relationships between dimensions and their significance in mathematical and physical equations. In summary, this work aims to demystify dimensional analysis, presenting its applications, and revealing the insights it offers into the bedrock of the laws governing the universe.

An analysis of the perception of time is made in comparison to the three common spatial dimensions. Whereas time is commonly thought to be more abstract and mysterious than the spatial dimensions, analysis of the human sense-perception system, and Einstein's construction of time in his space-time model for Special Relativity, determined that time shares the same conceptual framework and abstract status as the three spatial dimensions. What remains a profound mystery is human conceptualization of space-time in consciousness, and in conscious perception of space-time entities.

A series of arguments, phenomenological, empirical, and theological, are presented in favor of multiple dimensions of space and, within this, a spatial (B-theory) understanding of Time.

2021

Unlike nineteenth-century Gothic fiction, which tends to fixate on the past, the haunted and the ghostly, early weird fiction probes the very boundaries of reality - the laws and limits of time, space and matter. Here, unimaginable terrors lurk in hitherto unknown mirror dimensions, calamities in ultra-space threaten to wipe clean all evidence of our universe and experiments in non-Euclidean geometry lead to sickening consequences. In twelve speculative tales of our universe's mathematics and physics gone awry, this new anthology presents an abundance of curiosities - and terrors - with stories from Jorge Luis Borges, Miriam Allen deFord, Frank Belknap Long and Algernon Blackwood