Plot one cycle of a common helix.

ParametricPlot3D[ {Cos[t], Sin[t], t}, {t, 0, 2*Pi} ]

Modify the helix to have many turns, and therefore show the “spring” form associated with helices. The coefficients of t inside the sine and cosine functions govern the frequency with which helix twists, with higher coefficients meaning more twists.

ParametricPlot3D[ {Cos[8t], Sin[8t], t}, {t, 0, 2*Pi} ]

An example of a 2 dimensional vector-valued function. Takes advantage of sine and cosine functions to produce loops or cycles, but gets a much more complicated shape by having different frequencies for different components.

ParametricPlot[ { 2*Cos[3t], Sin[5t] }, {t, 0, 2*Pi} ]

Multiply sine and cosine terms for t (or something depending on t) to get a 2 dimensional spiral.

ParametricPlot[ { t * Cos[8t], t * Sin[8t]}, {t, 0, 2*Pi}]

Experimenting with many of the above ideas in 3 dimensions, roughly inspired by the idea of a conical helix.

ParametricPlot3D[ { t * Cos[10t], t * Sin[5t], t}, {t,-2*Pi,2*Pi} ]