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Multiplication by 2 modulo n

Multiplication by 2 modulo n
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2016-12-09 04:20
Coordinate Geometry
This example demonstrates the geometric shape and symmetry of point sets generated by multiplication by 2 modulo n in a coordinate system. Core mathematical concepts include modular arithmetic, geometric transformations, symmetry, and sequences. GeoGebra commands and objects used in construction are Intersect (intersection), Point (point), Circle (circle), and Sequence (sequence). Suitable for high school mathematics teaching to understand the geometric meaning and transformation properties of modular operations.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(yAxis) → B
  3. Circle(A, B) → c
  4. Sequence(Text[n - j, (cos(((2 * π) * j / n)), sin(((2 * π) * j / n)))], j, 1, n) → list1
  5. Sequence(Segment[(cos(((2 * π) * j / n)), sin(((2 * π) * j / n))), (cos(((2 * π) * j / n))^(2) - sin(((2 * π) * j / n))^(2), ((2 * sin(((2 * π) * j / n))) * cos(((2 * π) * j / n))))], j, 0, n - 1) → a
  6. Sequence((cos(((2 * π) * j / n)), sin(((2 * π) * j / n))), j, 1, n) → list2

AI Drawing Prompt Reference

Draw a coordinate system with the x-axis and y-axis intersecting at the origin. Take a point B on the y-axis, and draw a circle c with A and B as its diameter. Then, select n points on the circle c with coordinates (cos((2πj/n), sin((2πj/n))), where j ranges from 1 to n. Finally, connect each point to its symmetric counterpart with respect to the origin to form a series of line segments.
The prompt above was compiled from the original drawing commands and is for reference only.