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Many of these problems are easily solvable provided that other geometric transformations are allowed; for example, neusis construction can be used to solve the former two problems. In terms of algebra , a length is constructible if and only if it represents a constructible number , and an angle is constructible if and only if its cosine is a ...
A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. [5] This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape.
An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high. [122]
A college student just solved a seemingly paradoxical math problem—and the answer came from an incredibly unlikely place. Skip to main content. 24/7 Help. For premium support please call: 800 ...
Real algebraic geometry is the study of real algebraic varieties. The fact that the field of the real numbers is an ordered field cannot be ignored in such a study. For example, the curve of equation x 2 + y 2 − a = 0 {\displaystyle x^{2}+y^{2}-a=0} is a circle if a > 0 {\displaystyle a>0} , but has no real points if a < 0 {\displaystyle a<0} .
Packing circles in simple bounded shapes is a common type of problem in recreational mathematics. The influence of the container walls is important, and hexagonal packing is generally not optimal for small numbers of circles. Specific problems of this type that have been studied include: Circle packing in a circle; Circle packing in a square
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics. [131] It has close connections to convex analysis, optimization and functional analysis and important applications in number theory. Convex geometry dates back to antiquity. [131]
These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube, a polyhedron that is by alternating a cube. This form has Coxeter diagram and Schläfli symbol h { 4 , 3 } {\displaystyle \mathrm {h} \{4,3\}} .