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Generalized Balanced Ternary
Generalized balanced ternary is a generalization of the balanced ternary numeral system to represent points in a higher-dimensional space. It was first described in 1982 by Laurie Gibson and Dean Lucas. It has since been used for various applications, including geospatial and high-performance scientific computing. General form Like standard positional numeral systems, generalized balanced ternary represents a point p as powers of a base B multiplied by digits d_i. p = d_0 + B d_1 + B^2 d_2 + \ldots Generalized balanced ternary uses a transformation matrix as its base B. Digits are vectors chosen from a finite subset \ of the underlying space. One dimension In one dimension, generalized balanced ternary is equivalent to standard balanced ternary, with three digits (0, 1, and −1). B is a 1\times 1 matrix, and the digits D_i are length-1 vectors, so they appear here without the extra brackets. \beginB &= 3 \\ D_0 &= 0 \\ D_1 &= 1 \\ D_2 &= -1\end Addition table This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Ternary
Balanced ternary is a ternary numeral system (i.e. base 3 with three Numerical digit, digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2. The balanced ternary system can represent all integers without using a separate minus sign; the value of the leading non-zero digit of a number has the sign of the number itself. The balanced ternary system is an example of a Non-standard positional numeral systems, non-standard positional numeral system. It was used in some early computers and has also been used to solve balance puzzles. Different sources use different glyphs to represent the three digits in balanced ternary. In this article, T (which resembles a typographical ligature, ligature of the minus sign and 1) represents −1, while 0 and 1 represent themselves. Other conventions include using '−' and '+ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numeral System
A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number ''eleven'' in the decimal or base-10 numeral system (today, the most common system globally), the number ''three'' in the binary or base-2 numeral system (used in modern computers), and the number ''two'' in the unary numeral system (used in tallying scores). The number the numeral represents is called its ''value''. Additionally, not all number systems can represent the same set of numbers; for example, Roman, Greek, and Egyptian numerals don't have a representation of the number zero. Ideally, a numeral system will: *Represent a useful set of numbers (e.g. all integers, or rational numbers) *Give every number represented a unique representation (or a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positional Numeral System
Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred (however, the values may be modified when combined). In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string. The Babylonian numeral system, base 60, was the first positional system to be developed, and its influence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transformation Matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then there exists an m \times n matrix A, called the transformation matrix of T, such that: T( \mathbf x ) = A \mathbf x Note that A has m rows and n columns, whereas the transformation T is from \mathbb^n to \mathbb^m. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors. Uses Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation. This also allows transformations to be composed easily (by multiplying their matrices). Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on an n-dimensional Euclidean space R''n'' can be represented as linear transformations on the ''n''+1-dimensional space R''n''+1. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A '' vector quantity'' is a vector-valued physical quantity, including units of measurement and possibly a support, formulated as a '' directed line segment''. A vector is frequently depicted graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \stackrel \longrightarrow. A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word means 'carrier'. It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Visualization Of Three-digit 2D Generalized Balanced Ternary Numbers
Visualization or visualisation may refer to: *Visualization (graphics), the physical or imagining creation of images, diagrams, or animations to communicate a message * Data and information visualization, the practice of creating visual representations of complex data and information * Music visualization, animated imagery based on a piece of music * Mental image, the experience of images without the relevant external stimuli * "Visualization", a song by Blank Banshee on the 2012 album ''Blank Banshee 0'' See also * Creative visualization (other) * Visualizer (other) * * * * Graphics * List of graphical methods, various forms of visualization * Guided imagery, a mind-body intervention by a trained practitioner * Illustration, a decoration, interpretation or visual explanation of a text, concept or process * Image, an artifact that depicts visual perception, such as a photograph or other picture * Infographics Infographics (a clipped compound of "informatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosper Curve
The Gosper curve, named after Bill Gosper, also known as the Peano-Gosper Curve and the flowsnake (a spoonerism of Koch snowflake, snowflake), is a space-filling curve whose limit set is rep-tile, rep-7. It is a fractal curve similar in its construction to the dragon curve and the Hilbert curve. The Gosper curve can also be used for efficient hierarchical hexagonal clustering and indexing. Lindenmayer system The Gosper curve can be represented using an L-system with rules as follows: * Angle: 60° * Axiom: A * Replacement rules: ** A \mapsto A-B--B+A++AA+B- ** B \mapsto +A-BB--B-A++A+B In this case both A and B mean to move forward, + means to turn left 60 degrees and - means to turn right 60 degrees - using a "turtle"-style program such as Logo programming language, Logo. Properties The space filled by the curve is called the Gosper island. The first few iterations of it are shown below: The Gosper Island can tessellation, tile the Plane (mathematics), plane. In fact, se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
