11.6 Summary
Key Points
- Fractals are self-similar shapes, they repeat one pattern at smaller and smaller scales. They show up in nature (ferns, rivers, coastlines) and in art (Khajuraho temples, Fulani blankets, Escher prints).
- The Sierpinski Carpet keeps 8 of 9 squares each stage: R_n = 8^n survive, holes grow by H_{n+1} = H_n + R_n, and the area shrinks as \left(\tfrac{8}{9}\right)^n \to 0.
- The Sierpinski Triangle keeps 3 of 4 triangles each stage: T_n = 3^n survive and the area shrinks as \left(\tfrac{3}{4}\right)^n \to 0.
- The Koch Snowflake has 3\times 4^n sides at stage n and perimeter (side s) 3s\times\left(\tfrac{4}{3}\right)^n, a finite area with an infinite perimeter.
- The Cantor Set keeps 2 of every 3 pieces of length each stage: 2^n segments survive with total length \left(\tfrac{2}{3}\right)^n \to 0, a fractal on a line.
- A solid’s profile depends on the viewpoint; one profile cannot identify a solid, but three (front, top, side) usually can.
- A prism (n-gon) has n+2 faces, 2n vertices, 3n edges; a pyramid (n-gon base) has n+1 faces, n+1 vertices, 2n edges. All obey Euler’s formula F + V = E + 2.
- A net is an unfolded solid: a cube and an octahedron each have 11 nets, a regular tetrahedron has 2, a dodecahedron has 43,380, and a sphere has none.
- The shortest path on a cuboid’s surface is the straight line on the correct net; the way you unfold matters.
- Projections record a solid on a plane; a projected length satisfies p \le l (equal only when the line is parallel to the plane), and parallel lines stay parallel.
- Balancing a cube on a corner gives its isometric projection (a regular hexagon), in which all edges project to equal lengths, the basis of isometric grid drawing.