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(That is, merge the two boundary circles so they become just one circle.) To form the connected sum of more than two surfaces, successively take the connected sum of two of them at a time until they are all connected. In this sense, a genus g surface resembles the surface of g doughnuts stuck together side by side, or a 2-sphere with g handles ...
One such extension is to find the maximum possible density of a system with two specific sizes of circle (a binary system). Only nine particular radius ratios permit compact packing , which is when every pair of circles in contact is in mutual contact with two other circles (when line segments are drawn from contacting circle-center to circle ...
Mathematical analysis of 2D packing of circles (2022). H C Rajpoot from arXiv "The best known packings of equal circles in a circle (complete up to N = 2600)" "Online calculator for "How many circles can you get in order to minimize the waste?"
A circle containing one acre is cut by another whose center is on the circumference of the given circle, and the area common to both is one-half acre. Find the radius of the cutting circle. The solutions in both cases are non-trivial but yield to straightforward application of trigonometry, analytical geometry or integral calculus.
Circle packing in a square is a packing problem in recreational mathematics, where the aim is to pack n unit circles into the smallest possible square.Equivalently, the problem is to arrange n points in a unit square aiming to get the greatest minimal separation, d n, between points. [1]
A concrete model consists of two unit circles in perpendicular planes, each passing through the center of the other. [2] This model minimizes the ropelength of the link and until 2002 the Hopf link was the only link whose ropelength was known. [4] The convex hull of these two circles forms a shape called an oloid. [5]
But after each attempt, nothing. No usable DNA profile ever came back. So he was starting from scratch. Meantime, Mollet decided to turn up the pressure.
A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the relevant sort. For instance: The sphere and a disc both have genus zero. A torus has genus one, as does the surface of a coffee mug with a handle. This is the source of the joke "topologists are people who can't tell their donut from their coffee ...