Youth Program · Intermediate

Approaching Euler’s number

Starting with repeated products such as (1½)², (1⅓)³, and (1.01)¹⁰⁰ led to the general expression (1 + 1/n)ⁿ. As n grows, this expression converges to e. Reversing the relationship gives the useful large-n approximation e^(1/n) ≈ 1 + 1/n.

Exponents, logarithms, and music

Fractional exponents were interpreted as roots: for example, 2^(1/12) is the factor that must be multiplied twelve times to double a frequency. This is the basis of equal-tempered tuning, where twelve equal-ratio half steps take 440 Hz to 880 Hz. Logarithms answer the inverse question—log₂(1000) is the number of doublings needed to multiply a frequency by 1,000, while the natural logarithm is log base e.

• Human hearing example: From roughly 20 Hz to 20,000 Hz is a factor of 1,000, or about ten octaves because log₂(1000) ≈ 10.
• Musical interval factor: Each equal-tempered half step multiplies frequency by 2^(1/12), approximately 1.0595.

Tangents and numerical approximation

A tangent slope can be approximated using rise over run across an increasingly small change. This led to Newton’s method: begin with a guess, use the tangent line to obtain a better guess, and repeat until the estimates converge. The same iterative idea is used to calculate roots numerically.

Explore Newton’s method

Study Newton’s method and try implementing it in Java, JavaScript, or Python.

• Core task: Write a program that approximates an nth root of a number, or begin with the square root of 2.
• Extension: Create a graphic or animation showing each tangent-line step converging toward the root.

Euler’s characteristic and the Platonic solids

For any polyhedral mesh topologically equivalent to a sphere, F − E + V = 2. Shared features must not be double-counted: for an icosahedron, 20 triangular faces contribute 20 × 3 edge appearances, but every edge is shared by two faces, giving 30 edges; its 60 vertex appearances are shared five at a time, giving 12 vertices.

• Tetrahedron: Schläfli symbol {3,3}; F = 4, E = 6, V = 4.
• Cube: Schläfli symbol {4,3}; F = 6, E = 12, V = 8.
• Octahedron: Schläfli symbol {3,4}; F = 8, E = 12, V = 6.
• Dodecahedron: Schläfli symbol {5,3}; F = 12, E = 30, V = 20.
• Icosahedron: Schläfli symbol {3,5}; F = 20, E = 30, V = 12.

Duality, geodesics, and topology

A dual is formed by connecting the centers of a solid’s faces: the cube and octahedron are duals, the dodecahedron and icosahedron are duals, and the tetrahedron is self-dual. Geodesic subdivisions produce mostly hexagons while preserving 12 pentagonal locations; related structures occur in C60 fullerene molecules and some viruses.

• Topological equivalence: Shapes are equivalent when one can be continuously deformed into the other without tearing or creating a hole.
• Sphere versus torus: Sphere-like meshes have Euler characteristic 2, while a torus or mug-like surface has Euler characteristic 0.
• Question to contemplate: Why does F − E + V remain invariant as a sphere-like mesh is subdivided or deformed?

Seeing 45° trigonometric values

A square of area 1/2 has side length √(1/2) = √2/2. This makes sin 45° and cos 45° visually equal to √(1/2), while sec 45° and csc 45° equal √2; quadrant position determines the signs. At 225°, for example, sine and cosine are both −√(1/2), secant and cosecant are both −√2, and tangent and cotangent are both 1.

Study the five Platonic solids

Know each solid’s faces, edges, vertices, Schläfli symbol, Euler-characteristic calculation, and dual partner. Make a comparison chart, and also learn about Ludwig Schläfli and his role in the development of modern mathematics.

Place value works in every integer base

In base b, the available digits run from 0 through b−1, and each place represents a power of b. A number can therefore be expanded by multiplying each digit by its corresponding power of the base.

• Binary: 10₂ means 2, 11₂ means 3, and 100₂ means 4.
• Hexadecimal: A represents 10 and F represents 15. For example, 3A₁₆ = 3×16 + 10 = 58, while FF₁₆ = 15×16 + 15 = 255.
• Largest-digit shortcut: Repeated largest digits are one less than the next power of the base: FF₁₆ is one less than 100₁₆, and FFF₁₆ is one less than 1000₁₆, so FFF₁₆ = 4095.

Hexadecimal colors and Processing

Hexadecimal color codes contain three two-digit numbers describing the brightness of red, green, and blue. Each channel ranges from 00 to FF, equivalent to 0 through 255 in base 10.

• Additive light: Full red, green, and blue light combine to make white. This differs from mixing pigments, which absorb more light and become darker.
• Color experiments: Processing was used to vary RGB values, create colors such as yellow, magenta, and cyan, and connect mouse position to changing color values.
• Coding basics: The group examined screen dimensions, setup and draw functions, mouseX and mouseY, and return types such as void, int, float, char, and boolean.

Why change bases?

Familiar questions can trigger memorized answers without fresh reasoning. Changing the base makes familiar arithmetic unfamiliar again, encouraging “beginner’s mind,” adaptability, and active mathematical thought. Meditation was suggested as one way to quiet automatic reactions and approach familiar subjects with greater attention.

Optional color-picker challenge

Create a hexadecimal color picker in JavaScript or Python. Map mouse position to RGB values, color the display accordingly, and show the corresponding hexadecimal code. Beginners can first download Processing and work through introductory coding exercises.

Creative-coding introduction

The Coding Train, taught by Daniel Shiffman, was recommended for its friendly beginner lessons. Use its “Get Started” material for an extended introduction to Processing and creative coding.

Coordinates on the unit circle

For an angle θ on a circle of radius 1, the point on the circle is (cos θ, sin θ). Cosine gives the horizontal x-coordinate, while sine gives the vertical y-coordinate; both outputs remain between −1 and 1.

• Sine landmarks: sin(0°)=0, sin(90°)=1, sin(180°)=0, sin(270°)=−1, and sin(360°)=0.
• Cosine landmarks: cos(90°)=0 and cos(180°)=−1.
• Pythagorean connection: Because the radius is 1, the right triangle in the circle gives sin²(θ)+cos²(θ)=1.

From circles to waves and vectors

Plotting the changing vertical coordinate as the angle advances produces a sine wave; cosine produces the same basic wave shifted by 90°. The pair (cos θ, sin θ) also specifies a direction, which is why these functions are useful for vectors, rotations, programming, and smooth motion.

• Inputs and outputs: The input can cover any angle, while sine’s output stays between −1 and 1.
• Higher dimensions: The class briefly used a double helix and a rotating hypercube to discuss projections, cross-sections, and how sine and cosine help describe rotations.

The six trigonometric functions

The diagram also introduced tangent, cotangent, secant, and cosecant as visible segments related to the unit circle. Examples included tan(45°)=1, sec(45°)=√2, cot(45°)=1, and the identity 1+tan²(θ)=sec²(θ).

Practice for the week

Use the unit-circle diagram to identify right triangles and create Pythagorean equations relating sine, cosine, tangent, secant, cotangent, and cosecant. Continue practicing variable-base counting; Vladimir was also asked to look up point-slope form as another representation of a line.

Arduino game project

Vladimir shared an Arduino build using an LED strip, capacitor, wires, connectors, and a button box. The plan is to turn it into a one-dimensional video game combining electronics, programming, and mathematics.

Multiplication as visible area

A product can be seen as the number of unit squares in a rectangle. Breaking the rectangle into hundreds, tens, and ones made larger products directly visible and supported mental calculation.

• 11²: 100 + 10 + 10 + 1 = 121.
• 22²: 400 + 40 + 40 + 4 = 484.
• 101²: 10,000 + 100 + 100 + 1 = 10,201.
• Beyond squares: The same area method gave products such as 21 × 22 = 462 and 13 × 14 = 182.

Place value depends on the base

In base b, each position represents a power of b rather than a power of 10. Changing the base does not change the quantity—only the symbols and place-value groupings used to describe it.

• Base 3: Counting proceeds 0, 1, 2, 10, 11, 12, 20, 21, 22, 100; 100₃ represents 9 in base 10.
• Base 8: 40₈ = 32₁₀ and 41₈ = 33₁₀.
• Base 9: The base-10 quantity 11 is written 12₉, and its square is 144₉.
• Base 12 fractions: 1/2 = 0.6₁₂, 1/3 = 0.4₁₂, 1/4 = 0.3₁₂, and 1/6 = 0.2₁₂. The expansion of 1/5 repeats as 0.24972497…₁₂.

Clarity rather than competition

Students were encouraged to slow down, visualize, and even close their eyes while waiting for an answer to become clear. Mathematics was presented as an experience of beauty and mental clarity, with everyone taking satisfaction in one another’s success.

Practice visualizing squares

Practice square numbers during the week by mentally building the area diagrams and counting their hundreds, tens, and ones. Suggested examples from class include 13² = 169, 21² = 441, 22² = 484, 31² = 961, and 41² = 1,681.

Can a base system change from place to place?

An open question was raised about whether a positional system could use an asymmetrical or changing base rather than one fixed base. No conclusion was reached; it remains an idea for later investigation.