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An algebraic surface of degree 7. The Labs septic is an example of a septic surface.

A space curve consisting of a spiral wound around a helix. It has parametric equations x = [R+acos(omegat)]cost (1) y = [R+acos(omegat)]sint (2) z = ht+asin(omegat). (3)

Let X and Y be topological spaces. Then their join is the factor space X*Y=(X×Y×I)/∼, (1) where ∼ is the equivalence relation (x,y,t)∼(x^',y^',t^')<=>{t=t^'=0 and x=x^'; or ; ...

A condition which, if true, guarantees that a result is also true. (However, the result may also be true if the condition is not met.) If a condition is both necessary and ...

Let the speed sigma of a closed curve on the unit sphere S^2 never vanish. Then the tangent indicatrix, also called the tantrix, tau=(sigma^.)/(|sigma^.|) is another closed ...

Let x be a point in an n-dimensional compact manifold M, and attach at x a copy of R^n tangential to M. The resulting structure is called the tangent space of M at x and is ...

An exchange of two elements of an ordered list with all others staying the same. A transposition is therefore a permutation of two elements. For example, the swapping of 2 ...

cos(pi/(10)) = 1/4sqrt(10+2sqrt(5)) (1) cos((3pi)/(10)) = 1/4sqrt(10-2sqrt(5)) (2) cot(pi/(10)) = sqrt(5+2sqrt(5)) (3) cot((3pi)/(10)) = sqrt(5-2sqrt(5)) (4) csc(pi/(10)) = ...

Construction of the angle pi/4=45 degrees produces an isosceles right triangle. Since the sides are equal, sin^2theta+cos^2theta=2sin^2theta=1, (1) so solving for ...

Two quantities a and b which are not equal are said to be unequal, and this relationship can be denoted a!=b.