1. **Patterns** Patterns are sequences or arrangements that repeat in a predictable way. In math, patterns help us find the next number or shape by recognizing the rule that generates the sequence. 2. **Isometries** Isometries are transformations that keep distances the same. The four types are: - Translation: sliding a shape without rotating or flipping it. - Rotation: turning a shape around a point. - Reflection: flipping a shape over a line. - Glide reflection: sliding then flipping. These transformations change position or orientation but not size or shape. 3. **Golden Ratio** The golden ratio $\phi$ satisfies the equation $$x^2 - x - 1 = 0$$ Solving this quadratic using the formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}$$ Since $\phi$ is positive, we take: $$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$$ This ratio appears in art, architecture, and nature because it is aesthetically pleasing. 4. **Fractals** Fractals are shapes that look similar at any zoom level (self-similar). They are created by repeating a simple process infinitely. Examples include the Mandelbrot set and natural forms like coastlines and clouds. Fractals show how complex patterns arise from simple rules.