majiayu000 Perception
Summary
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)
Key Primitives
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).
- computing vertical components of circular or oscillatory motion
- modeling periodic phenomena like waves and vibrations
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).
- computing horizontal components of circular or oscillatory motion
- finding phase relationships between periodic signals
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).
- any calculation involving continuous quantities
- measuring distances, areas, or physical quantities
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).
- defining trigonometric functions geometrically
- representing angles and rotations in the plane
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.
- computing the angle between two vectors
- finding the component of one vector along another direction
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.
- counting objects or elements in a set
- establishing the basis for mathematical induction
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.
- computing the length of the hypotenuse in a right triangle
- finding distance between two points in Euclidean space
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).
- representing quantities with both magnitude and phase
- solving polynomial equations that have no real roots
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.
- measuring distance from zero or between two numbers
- bounding the size of a quantity regardless of sign
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.
- converting between trigonometric and exponential forms of complex numbers
- simplifying products and powers of trigonometric expressions using exponentials
Composition Patterns
- Integers + perception-natural-numbers -> Complete additive group with identity and inverses (sequential)
- Rational Numbers + perception-integers -> A number system where division is always defined (except by zero) (sequential)
- Irrational Numbers + perception-rational-numbers -> The complete real number line without gaps (parallel)
- Real Numbers + perception-real-line-completeness -> A number system where limits of convergent sequences always exist (nested)
- Complex Numbers + perception-unit-circle -> Polar form of complex numbers: z = re^(itheta) (parallel)
- Absolute Value + perception-number-line -> Distance between two points on the line: |a - b| (sequential)
- Number Line + perception-absolute-value -> Metric space structure on R with distance d(a,b) = |a-b| (parallel)
- Density of Rationals + perception-real-line-completeness -> Understanding that Q is dense but not complete — R fills the gaps (parallel)
- Triangle Inequality for Absolute Value + perception-distance-formula -> Metric space axiom: d(a,c) <= d(a,b) + d(b,c) (sequential)
- Unit Circle + perception-complex-numbers -> Complex numbers of modulus 1: z = e^(i*theta) on the unit circle (parallel)
Cross-Domain Links
- waves: Compatible domain for composition and cross-referencing
- change: Compatible domain for composition and cross-referencing
- structure: Compatible domain for composition and cross-referencing
- synthesis: Compatible domain for composition and cross-referencing
Activation Patterns
- number
- count
- distance
- magnitude
- circle
- trigonometric
- angle
- inner product
- orthogonal