Measurement and Geometry. Geometry Formulas. 3D Shapes . Prisms. GeoGebra. Knot theory. In this Section we will cover the following aspects: Table of contents 1.Motivation ? 2.Basic knot theory ? 2.1.What is a mathematical knot? ? 2.2.Equivalence/deformation of knots ? 2.4.Diagrams of knots and links ? 2.5.3.Amphicheiral knots ?
2.5.4.Alternating knots ? 3.Advanced knot theory ? 3.1.Reidemeister moves and Reidemeister theorem ? 3.2.Invariants of knots and links ? 3.2.1.numeric invariants: unknotting number and crossing number ? 3.2.3.polynomial invariants ? 3.3.Tabulation of knots and links ? 4.Modern aspects of knot theory ? 5.References on knot theory that I find useful ? 1.Motivation My current research refers to the subject of my PhD Thesis, focused on "Symbolic numeric algorithms for genus computation of plane complex algebraic curves based on knot theory". To give a basic summary of the most important aspects of knot theory which we use for the work of the current project; to show how knot theory can be analyzed from other important significant domains of mathematics; A knot or . In. Heighway Dragon. Typesetting math: 16% Classic Iterated Function Systems <p style="margin-left:20%; margin-top:10px"><b>Note</b>: This site uses Javascript.
It also uses <a href=" target="_blank">MathJax</a> (which requires Javascript) to display mathematical expressions. Please turn on JavaScript, then refresh the page. </p> Description ConstructionAnimation Construction via line segments Begin with a line segment. Construction via paper folds Here is how Heighway originally constructed the dragon. ConstructionAnimation Construction via triangles For yet another way to construct the Heighway dragon, start with an isosceles right triangle H0. The Heighway dragon is the limiting set of this iterative construction. IteratedFunctionSystem To think of this construction as a result of affine transformations, consider the first iteration of the line segment from (0,0) to (1,0) as shown by the two red line segments in the following figure.
IFSAnimation where {\bf{r}} = \frac{1}{{\sqrt 2 }}. 9 40 - Knot Atlas. Knots and Surfaces. Surfaces With Boundary In class, we have mainly studied surfaces without boundary - the sphere, torus, klein bottle, etc. We have been particularly interested in the Euler characteristics of these surfaces. How do we find the Euler characteristic of a surface with boundary?
Any surface with boundary can be "capped off" to create a surface without boundary simply by filling each boundary component with a disk. Since a disk has Euler characteristic 1, the Euler characteristic of a surface without boundary is equal to the Euler characteristic of the surface obtained by capping it off, minus the number of boundary components in the original surface. The Euler characteristic of a surface with boundary does not uniquely specify the surface. That is, two non-homeomorphic surfaces may have the same Euler characteristic. So what is different about these surfaces? The surface's Euler characteristic The number of boundary components of the surface Whether or not the surface is orientable. Algebraic Curves. Minimal Surfaces. Riemann sphere. The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection – details are given below).
In mathematics, the Riemann sphere, named after the 19th century mathematician Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers. In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds.
Extended complex numbers[edit] The extended complex numbers consist of the complex numbers C together with ∞. Geometrically, the set of extended complex numbers is referred to as the Riemann sphere (or extended complex plane). Arithmetic operations[edit] Rational functions[edit] Metric[edit] The Scientific Graphics Project. Geometry: Requirements for a Visualization System for 2020. In this essay, i give a list of requirements that i think is necessary for a software system for creating scientific visualization for the next decade.
For the past 10 years, i have been strongly interested in mathematical visualization. I'm not a professor, and am not doing it for the educational purposes per se. Geometry has just been beautiful for me, and i'm also a professional programer. Programing computers to visualize geometric theorems or geometric objects, has been a immense personal pleasure. A particular aspect of visualization, is to design it so that when viewed, it forces a immediate and crystal clear understanding of the object or theorem. In the following, i give itemized descriptions on features necessary in a visualization system, based on my experiences. Requirement: Real-Time 3D Rotation For math visualization, it is absolutely necessary to be able to rotate the object in real-time.
Requirement: Real-time Interactivity This real-time interactivity is frequently needed. 25 Gifs That Teach You Math Concepts Better Than Your Teacher Did. Using units of measurement. Shape. Geometric Reasoning. Location & Transformation. Shape. Skillswise - Maths. The Sound of Silence II. By Thomas Váczy Hightower Standing waves In the first part of The Sound of Silence we have mainly investigated the broaden concept of motion, the pendulum and its strange behavior at the quantum level. Now we will explore other meta physical aspects of music and sounds.
Standing waves is an essential phenomenon in the creation of the musical tone. The key word in standing waves is order. De Borglie waves On the quantum level sound does not have meaning, but standing wave does still perform order, when we are looking at the orbiting electrons in an atom. The orbit is only stable if it meets the condition for standing waves. Table of the Elements The periodic tables of Elements reflexes the wave pattern of atoms. This way of viewing the table of elements as a square of octaves does not have a scientific value. In accordance with the cosmogony and the theory of the Big Bang for about 13,5 billion years ago, only Hydrogen and Helium existed in the new born Universe.
The concept of harmony. CARBON ATOMS, MELANIN AND SACRED GEOMETRY. After 400 Years, Mathematicians Find A New Class Of Shapes. Skillswise - Maths. The Bizarre Object We Believed Was Impossible to Visualize. Well I wouldn't suggest you chop your fingers off like Django. But the rest of it yes I suggest you try to write. Also I as a Humanities major don't really understand this but it does sound cool. I'm an engineering major, and I still don't get it. Maybe if it was written in my first language... Well I assume at some point the article will be translated or some popular science magazine will explain it all in your native tongue. But basically, if I understand it correctly, what's being described here is something like the toridal equivalent of a pseudosphere. Well, in reply to the losing fingers in a fire example, I could say that I'm not expecting people to take up complex variable analysis of poles and residues as a hobby but— They can at least try to read some articles of these interesting discoveries in math written for general audiences.
Just like your suggestion that all people should at least try their hand at writing, if only to expand and work their brains. Measuring Angles & Types of Angles. Symmetry Unit. Transformations. 3D Shapes . Prisms. Logic - the fractal or scale-free middle of a hierarchy. What is a fractal? And why does it matter? A fractal shape is geometry with scale – a middle ground or internal axis of scale symmetry. So it is no surprise that fractals should turn out to model an organic world which is based on a logic of growth and hierarchical scale. Ordinary geometry – like ordinary logic or ordinary science – does not take account of scale. Differences in size are not seen as something fundamental. We could say the same for a propeller, a whorl of turbulence in a stream, or many other geometrical forms.
Fractals are also Platonic in being represented as static, ready-made, mathematical objects. But let’s start first by considering some classic fractals shapes such as the Koch curve, Cantor set, Sierpinski gasket and Menger sponge. So with the Koch curve, we begin with a flat line segment – the initiating state. Note several things here. Originally the 1D line just sat there in the 2D void. In theory a Koch curve can keep approaching this limit forever.