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Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e. a non-orientable surface of (non-orientable) genus n). (This number is also called the demigenus.)
It allows to link multiple representations dynamically. For example, changing a formula can instantly change the graph, the table of values, and the text read-out for the function represented in all these ways. Technology use can increase accuracy and speed of data collection and allow real-time visualization and experimentation. [14]
The Mandelbrot set, one of the most famous examples of mathematical visualization.. Mathematical phenomena can be understood and explored via visualization.Classically, this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century).
The edges and vertices of these six regions form Tietze's graph, which is a dual graph on this surface for the six-vertex complete graph but cannot be drawn without crossings on a plane. Another family of graphs that can be embedded on the Möbius strip, but not on the plane, are the Möbius ladders , the boundaries of subdivisions of the ...
A toroidal graph that cannot be embedded in a plane is said to have genus 1. The Heawood graph, the complete graph K 7 (and hence K 5 and K 6), the Petersen graph (and hence the complete bipartite graph K 3,3, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks, [1] and all Möbius ladders are toroidal.
While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a specific vertex. [ 3 ] It is unknown whether graph coloring and the maximum independent set problem can be solved in polynomial time on even-hole-free graphs, or whether they are NP-complete.
One of Nathan's videos — showing him successfully calculating a lengthy equation that includes multiplication, addition, subtraction, division and even square root — gained 40,000 views ...
Animation of the missing square puzzle, showing the two arrangements of the pieces and the "missing" square Both "total triangles" are in a perfect 13×5 grid; and both the "component triangles", the blue in a 5×2 grid and the red in an 8×3 grid.
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