The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection. The main uses of this term are fivefold: 1. * Apollonius showed that a circle can be defined as the set of points in a plane that have a specified ratio of distances to two fixed points, known as foci. This Apollonian circle is the basis of the Apollonius pursuit problem. It is a particular case of the first family described in #2. 2. * The Apollonian circles are two families of mutually orthogonal circles. The first family consists of the circles with all possible distance ratios to two fixed foci (the same circles as in #1), whereas the second family consists of all possible circles that pass through both foci. These circles form the basis of bipolar coordinates. 3. * The circles of Apollonius of a triangle are three circles, each of which passes through one vertex of the triangle and maintains a constant ratio of distances to the other two. The isodynamic points and Lemoine line of a triangle can be solved using these circles of Apollonius. 4. * Apollonius' problem is to construct circles that are simultaneously tangent to three specified circles. The solutions to this problem are sometimes called the circles of Apollonius. 5. * The Apollonian gasket—one of the first fractals ever described—is a set of mutually tangent circles, formed by solving Apollonius' problem iteratively. (Wikipedia).
Apollonius' circle construction problems | Famous Math Problems 3 | NJ Wildberger
Around 200 B.C., Apollonius of Perga, the greatest geometer of all time, gave a series of related problems; how to construct a circle in the plane touching three objects, where the objects are either a point (P), a line (L) and or a circle (C). Many mathematicians have studied this most fa
From playlist Famous Math Problems
Quickly fill in the unit circle by understanding reference angles and quadrants
👉 Learn about the unit circle. A unit circle is a circle which radius is 1 and is centered at the origin in the cartesian coordinate system. To construct the unit circle we take note of the points where the unit circle intersects the x- and the y- axis. The points of intersection are (1, 0
From playlist Trigonometric Functions and The Unit Circle
Studying Apollonian Circle Packings using Group Theory
This video is in response to a colleague who asked for short videos about how we use group theory in our research. One of my interests is Apollonian circle packings. Some links: Wikipedia: https://en.wikipedia.org/wiki/Apollonian_gasket An article in New Scientist by Dana Mackenzie: ht
From playlist Joy of Mathematics
How to Draw Tangent Circles using Cones
Solving the Problem of Apollonius with Conic Sections This video describes a non-standard way of finding tangent circles to a given set of 3 circles, known as the Problem of Apollonius. It uses conic sections rather than straightedge and compass. I feel this approach is more intuitive and
From playlist Summer of Math Exposition Youtube Videos
Apollonian packings and the quintessential thin group - Elena Fuchs
Speaker: Elena Fuchs (UIUC) Title: Apollonian packings and the quintessential thin group Abstract: In this talk we introduce the Apollonian group, sometimes coined the “quintessential” thin group, which is the underlying symmetry group of Apollonian circle packings. We review some of the e
From playlist My Collaborators
The Three/Four bridge and Apollonius duality for conics | Six: A course in pure maths 5 | Wild Egg
The Three / Four bridge plays an important role in understanding the remarkable duality discover by Apollonius between points and lines in the plane once a conic is specified. This is a purely projective construction that works for ellipses, and their special case of a circle, for parabola
From playlist Six: An elementary course in Pure Mathematics
Problem of Apollonius - what does it teach us about problem solving?
This video uses the problem of Apollonius as a way to introduce circle inversion and an important problem-solving technique - transforming a hard problem into a simpler one; then solve for the simpler, transformed version of the problem before doing the inverse transformation so that we ob
From playlist Geometry Gem
Circles are Infinity-Sided Polygons
It's natural to say that a circle is a polygon with infinite sides, so many that they become round and pi pops out. In this video, we explore this notion and how we can prove it.
From playlist Fun
Apollonius and polarity | Universal Hyperbolic Geometry 1 | NJ Wildberger
This is the start of a new course on hyperbolic geometry that features a revolutionary simplifed approach to the subject, framing it in terms of classical projective geometry and the study of a distinguished circle. This subject will be called Universal Hyperbolic Geometry, as it extends t
From playlist Universal Hyperbolic Geometry
Gary Antonick - Projectile on an Incline-No Calculation - G4G13 Apr 2018
Draw the path of a projectile on an incline. No calculation needed. A projectile bounces up and back down an inclined plane. How might this bounce path be drawn without calculation? This video shares a simple geometric technique (circles and lines) for producing a plane and launch angle t
From playlist G4G13 Videos
Curves from Antiquity | Algebraic Calculus One | Wild Egg
We begin a discussion of curves, which are central objects in calculus. There are different kinds of curves, coming from geometric constructions as well as physical or mechanical motions. In this video we look at classical curves that go back to antiquity, such as prominently the conic sec
From playlist Algebraic Calculus One from Wild Egg
👉 Learn about the unit circle. A unit circle is a circle which radius is 1 and is centered at the origin in the cartesian coordinate system. To construct the unit circle we take note of the points where the unit circle intersects the x- and the y- axis. The points of intersection are (1, 0
From playlist Evaluate Trigonometric Functions With The Unit Circle (ALG2)
Determine the point on the unit circle for an angle
👉 Learn how to find the point on the unit circle given the angle of the point. A unit circle is a circle whose radius is 1. Given an angle in radians, to find the coordinate of points on the unit circle made by the given angle with the x-axis at the center of the unit circle, we plot the a
From playlist Evaluate Trigonometric Functions With The Unit Circle (ALG2)
Alex Kontorovich - Numbers and Fractals [2017]
It is a very good time to be a mathematician. This millennium, while only a teenager, has already seen spectacular breakthroughs on problems like the Poincar´e Conjecture (solved by Grisha Perelman, who declined both a Fields Medal and a million dollar Clay Prize) and the near-resolution o
From playlist Mathematics
Determining where a point is on the unit circle
👉 Learn how to find the point on the unit circle given the angle of the point. A unit circle is a circle whose radius is 1. Given an angle in radians, to find the coordinate of points on the unit circle made by the given angle with the x-axis at the center of the unit circle, we plot the a
From playlist Evaluate Trigonometric Functions With The Unit Circle (ALG2)
STPM - Local to Global Phenomena in Deficient Groups - Elena Fuchs
Elena Fuchs Institute for Advanced Study September 21, 2010 For more videos, visit http://video.ias.edu
From playlist Mathematics
Tangents to Parametric Curves | Algebraic Calculus One | Wild Egg
Tangents are an essential part of the differential calculus. Here we introduce these important lines which approximate curves at points in an algebraic fashion -- finessing the need for infinite processes to support takings of limits. We begin by a discussion of tangents historically, espe
From playlist Algebraic Calculus One from Wild Egg
How to determine the point on the unit circle given an angle
👉 Learn how to find the point on the unit circle given the angle of the point. A unit circle is a circle whose radius is 1. Given an angle in radians, to find the coordinate of points on the unit circle made by the given angle with the x-axis at the center of the unit circle, we plot the a
From playlist Evaluate Trigonometric Functions With The Unit Circle (ALG2)
Ciro Ciliberto, Enumeration in geometry - 15 Novembre 2017
https://www.sns.it/eventi/enumeration-geometry Colloqui della Classe di Scienze Ciro Ciliberto, Università di Roma “Tor Vergata” Enumeration in geometry Abstract: Enumeration of geometric objects verifying some specific properties is an old and venerable subject. In this talk I will
From playlist Colloqui della Classe di Scienze