Foundational measurements and relationships. The axioms of the system — numbers, distance, angles, and the geometry of seeing.
Perception (Part I: Seeing) Chapters: 1, 2, 3 Plane Position: (-0.2, 0.2) radius 0.4 Primitives: 43
Foundational measurements and relationships. The axioms of the system — numbers, distance, angles, and the geometry of seeing.
Key Concepts: Sine Function, Cosine Function, Real Numbers, Unit Circle, Inner Product (Dot Product)
Sine Function (definition): The sine function sin: R -> [-1,1] is defined as the y-coordinate of the point on the unit circle at angle theta from the positive x-axis. It is periodic with period 2*pi, odd: sin(-theta) = -sin(theta).
Cosine Function (definition): The cosine function cos: R -> [-1,1] is defined as the x-coordinate of the point on the unit circle at angle theta. It is periodic with period 2*pi, even: cos(-theta) = cos(theta). cos(theta) = sin(theta + pi/2).
Real Numbers (definition): The real numbers R form a complete ordered field: closed under +, -, *, /, ordered by <, and satisfying the completeness axiom. R = Q union (R \ Q).
Unit Circle (definition): The unit circle is the set of points (x, y) in R^2 satisfying x^2 + y^2 = 1. Every point on the unit circle can be written as (cos(theta), sin(theta)) for a unique angle theta in [0, 2*pi).
Inner Product (Dot Product) (definition): The inner product (dot product) of vectors u = (u1,...,un) and v = (v1,...,vn) in R^n is u . v = sum_i u_i * v_i = |u||v|cos(theta), where theta is the angle between u and v.
Natural Numbers (axiom): The natural numbers N = {1, 2, 3, ...} satisfy the Peano axioms: there exists a first element 1, every element n has a unique successor S(n), no two elements share a successor, and the induction principle holds.
Pythagorean Theorem (theorem): In a right triangle with legs a and b and hypotenuse c: a^2 + b^2 = c^2. Conversely, if a^2 + b^2 = c^2 for a triangle with sides a, b, c, then the triangle is right-angled.
Complex Numbers (definition): The complex numbers C = {a + bi : a, b in R, i^2 = -1} form an algebraically closed field. Every complex number has modulus |z| = sqrt(a^2 + b^2) and argument arg(z) = atan2(b, a).
Absolute Value (definition): For x in R, the absolute value |x| = x if x >= 0, |x| = -x if x < 0. Equivalently, |x| = sqrt(x^2). It measures the distance from x to 0 on the number line.
Euler's Formula (identity): For all theta in R: e^(itheta) = cos(theta) + isin(theta). The special case theta = pi gives Euler's identity: e^(i*pi) + 1 = 0.