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One Point, Two Addresses

A study map for MAT 412 Test 1, Chapters 1 to 3. It runs in your browser and works offline.

Colour key
  • Blue is the Cartesian address: where z sits. Across by x, up by y. Slides and grid lines.
  • Orange is the polar address: how to reach z. Turn by θ, stretch by r. Rays and circles.
  • Green is the input: what you drag or draw in the z-plane.
  • Purple is the output: where f sends it, in the w-plane.
  • Dark to bright is time, from t = 0 to t = 1. Input and output run on one clock.
  • A solid arc is Arg z: the one angle we agreed to use, in (-π, π].
  • Faint arcs are the other members of arg z (θ ± 2π, θ ± 4π, …); the solid arc is Arg z. All of them belong to the set arg z.
  • A dashed grey ray is the seam: the negative real axis, where Arg jumps from π to -π.
  • The marks in the menu show which story threads pass through a place: T1 a blue disc (solid line), T2 an orange dashed ring (dashed line), T3 a purple diamond (dotted line).
  • Blue is a landing that stayed: settles or stays. On a pin it is the blue root.
  • Orange on a landing means the walk left. Orange on a pin means that root. On the Julia picture the handle for c is still orange; on the flat paper c is a hollow square, not a colour.
  • Purple is the third root, and only that.
  • Green is the plant: the start you drag.
  • Blue is a starting point whose walk stayed near 0. The solid region is the filled Julia set.
  • Orange is a starting point whose walk ran away. The orange handle is c, the slide added at every step. It is not a starting point.
  • Green is the starting point you drag. Its walk is drawn on top.
  • Blue is the blue root, and every guess Newton’s method sends there.
  • Orange is the orange root, and every guess it sends there.
  • Purple is the third root, and every guess it sends there. With two roots that handle is hidden.
  • Green is the guess you drag. Its steps are drawn on top. Black did not settle on a root.