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Heilbronn volumes
No full textEnsino Médio::MatemáticaThe general Heilbronn problem finds maxima for points in unit objects. In this Demonstration, the author has tried to maximize the smallest tetrahedral volume that can be formed from points in a unit cube. Likely none of these are optimal solution
Heilbronn volumes
No full textEnsino Médio::MatemáticaThe general Heilbronn problem finds maxima for points in unit objects. In this Demonstration, the author has tried to maximize the smallest tetrahedral volume that can be formed from points in a unit cube. Likely none of these are optimal solution
Heilbronn volumes
No full textEnsino Médio::MatemáticaThe general Heilbronn problem finds maxima for points in unit objects. In this Demonstration, the author has tried to maximize the smallest tetrahedral volume that can be formed from points in a unit cube. Likely none of these are optimal solution
Heilbronn volumes
No full textEnsino Médio::MatemáticaThe general Heilbronn problem finds maxima for points in unit objects. In this Demonstration, the author has tried to maximize the smallest tetrahedral volume that can be formed from points in a unit cube. Likely none of these are optimal solution
Ford circles
No full textEducação Superior::Ciências Exatas e da Terra::MatemáticaThe Farey sequence F(n) for a positive integer n is the set of irreducible rational numbers h/k with 0 ≤ h ≤ k ≤ n and (h,k)=1. For example, F(4) is 0/1, 1/4, 1/3, 1/2, 2/, 3/4 , 1/1. The Ford circle C(h,k) has radius 1/2k² and center (h/k , 1/2k²). It is tangent to the x axis at h/k and to the circles corresponding to the two neighbors of h/k in F(n
Limit laws
No full textAnalytic GeometryPairs of functions can be combined in six standard ways. The pairs are plotted as dashed curves and their combination as a solid brown line. The limit of the combination as x approaches 1 is the height of the red dotComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemátic
Ramsey(3,3) = 6
No full textnoneThe game of Sim, invented by Gustavus Simmons, matches Red against Blue on a hexagonal field of six dots. The players take turns drawing a line of their respective color between pairs of unconnected dots, losing if they make a triangle of their own color first.
This Demonstration shows all the 32768 2-colorings of the hexagon. When a set of vertices makes a triangle, the vertices are circled. All of the colorings contain at least one triangle.
The Ramsey problem R?(a,a) asks for the smallest n so that the complete graph Kn always contains a smaller monochromatic subgraph Ka, no matter how Kn is 2-colored. The graph that connects three points, K3, is a triangle. Since K5 can be 2-colored with no triangles (red star, blue pentagon), and since k6 always contains a triangle, the solution to the Ramsey problem R(3,3) is 6. The solution for R(4,4) is 18, with the 17-Paley graph and its inverse providing a 2-coloring for K17 without K4. The solution for R(5,5) is currently unknown, and it is predicted that the solution to R(6,6) will never be knownComponente Curricular::Ensino Fundamental::Séries Finais::Matemátic
Precision Error
No full textThe plots in this Demonstration show precision error. The formula arctan(x)+arctan(1/x)=π/2 is true for all real x≠0. To plot arctan(kx)+arctan(1/(kx))-π/2 for various k, Mathematica correctly finds numerical values very close to 0. In the plot, these infinitesimal deviations are magnified a quadrillion-foldComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemátic
Two Wheel Belt
No full textGeometryA classic trigonometry problem asks for the length of a belt wrapped around two wheels. The radii of the two wheels are x and y. The wheels are z units apartComponente Curricular::Ensino Médio::Matemátic
The eye-pentagon construction
No full textnoneWith a ruler and compass, an eye is easy to construct, leading to a regular pentagon construction. First draw a unit circle centered at O, the pupil. With centers at the top and bottom of the pupil, draw two circles of radius 2 to form the eyelids. Using the top of the pupil (A) as the center, draw an arc of radius 1 (AO) to intersect the upper eyelid at B. With C as the left corner of the eye, draw an arc with radius OC, and then a larger arc with radius BC. Draw a final arc with radius BC centered at O. Connect the intersection points for the pentagonComponente Curricular::Ensino Fundamental::Séries Finais::Matemátic
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