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Every term of the Taylor series of e^iθ is the previous term turned 90° and shrunk. Chain those perpendicular steps and a circle appears — an interactive lesson that builds rotation without assuming the unit circle. … Continue readingHow Straight Lines Make a Circle
Every large reorientation passes through a season where everything tastes like less. It is not a verdict on the turn — the walk to a new bearing swings to five times the circle before it settles, and the mid-turn reading is the one guaranteed to be wrong. … Continue readingThe Swing Wide
One condition at the origin — there shall be no undoing — makes the answer impossible to place and forces it to be walked to. A second dimension opens, ideation is born, and ideas turn out to be what a world invents when it cannot go back. … Continue readingThe Long Way Round
The cardinal ideas were never a curated list. Rotation is assembled from steps that do not rotate, the steps point in only four directions, and every idea you will ever have is a word spelled in that four-letter alphabet. … Continue readingFour Letters
Ideas do not arrive. Your bearing sweeps a circle of standing conditions, and what feels like a visitation is the geometry of the crossing — the same error a flatlander makes when a sphere passes through his plane. … Continue readingThe Flatlander’s Mistake