Vlad’s Mentorship

Seeing and constructing a hypercube

A drawing of a cube is a two-dimensional projection of a three-dimensional object; similarly, the displayed tesseract is a projection into 3D and then onto the screen. The construction pattern is point → line → square → cube → hypercube: copy the previous object in a new direction and connect corresponding vertices. A tesseract has 16 vertices, 32 edges, 24 square faces, and 8 cubic cells.

• Projection versus cross-section: A projection is a shadow-like image of the whole object; a cross-section is only the part where an object intersects a plane.
• Four-dimensional rotation: Using coordinates x, y, z, and w, rotations occur in six coordinate planes: xy, xz, yz, xw, yw, and zw.
• Higher-dimensional graphs: At high dimensions, the geometric picture becomes crowded, so it is useful to view the object as a graph of vertices and the edges connecting them.

Counting faces with combinations

The number of k-dimensional faces in an n-dimensional cube is 2^(n−k) × C(n,k), where k=0 counts vertices, k=1 edges, k=2 square faces, and k=3 cubic cells. The combination C(n,k)=n!/[k!(n−k)!] counts selections when order does not matter; this was explained by choosing students or books from a larger group. These combination values are also entries in Pascal’s triangle.

• 6D edges: 2^5 × C(6,1) = 32 × 6 = 192.
• 7D square faces: 2^5 × C(7,2) = 32 × (7×6÷2) = 672.
• 10D cubic cells: 2^7 × C(10,3) = 128 × (10×9×8÷(3×2×1)) = 15,360.
• Figurate-number connection: Triangular and tetrahedral numbers appear along diagonals of Pascal’s triangle; each sequence is formed by summing the preceding one.

Desmos combination construction

Vlad used a product expression and division by k! to recreate nCr in Desmos. The work showed why ordered selections must be divided by the k! possible reorderings to obtain combinations.

Compute n-cube counts directly

Practice calculating the number of edges, square faces, or cubic cells in an n-dimensional cube quickly using ordinary multiplication and division. Avoid relying on written factorial notation or the choose command; expand 2^(n−k) and C(n,k) directly, as in the 6D, 7D, and 10D examples.

Continue exploring the app library

Spend time independently manipulating the Hypercube Map, especially its projection, construction, perspective, cell-highlighting, and rotation controls. The Circle Area, number-sequence, and polyhedra explorations were also recommended for independent study.

Programming projects discussed

Vlad demonstrated a terminal-based AI coding workflow and his Cardputer schedule program, which displays classes, rooms, teachers, and times. He also showed a Pico Breadboard Kit Plus and discussed possible practical embedded-computing projects.

The angle argument for the five Platonic solids

For a regular n-gon, the exterior angle is 360°/n and each interior angle is 180° − 360°/n. At least three faces must meet at a vertex, and their angles must total less than 360° so the surface can fold into three dimensions.

• Triangles: Three, four, or five can meet at a vertex, producing the tetrahedron, octahedron, and icosahedron.
• Squares: Only three can meet, producing the cube; four squares total 360° and remain flat.
• Pentagons: Three meet with a 324° total, producing the dodecahedron.
• Hexagons and beyond: Three regular hexagons already total 360°, while larger polygons exceed it. Therefore no additional Platonic solids are possible.

Notation, truncation, and duality

A Schläfli symbol {p,q} records p sides per face and q faces meeting at each vertex. A vertex configuration lists the face sizes encountered around a vertex, which also works when several polygon types are present.

• Truncation: Cutting every corner of a cube gives vertex configuration 3.8.8; an octahedron gives 4.6.6; a tetrahedron gives 3.6.6; a dodecahedron gives 3.10.10; and an icosahedron gives 5.6.6—the soccer-ball form.
• Platonic duals: Connecting face centers reverses the Schläfli symbol: cube ↔ octahedron, dodecahedron ↔ icosahedron, while the tetrahedron is self-dual.
• Larger families: The 13 Archimedean solids are vertex-transitive and use at least two kinds of regular polygon. Their duals are the 13 Catalan solids, whose faces are congruent but generally not regular. The 92 Johnson solids use regular polygon faces but need not have identical vertices.

Program a polyhedron model in Processing

The suggested assignment was to model these solids in Processing using arrays. Store each vertex as XYZ coordinates, store each face as an ordered list of vertex indices, and loop through the face data to draw the form; a JSON file could eventually hold names, families, vertices, faces, and other metadata.

• Recommended starting point: Begin with one simple solid before attempting the full collection, and postpone camera controls until the geometry works.

Protect the mathematical thread

When an association or side question appears, first decide whether it advances the current idea. Save less relevant questions until the planned explanation is complete; your quick associations are a strength when they illuminate the mathematics rather than redirecting it.

Eternal Forms handout

Keep the saved PDF containing the lesson material on Platonic, Archimedean, Catalan, and Johnson solids and their historical development.

Question to revisit

What are the duals of the 92 Johnson solids called, and what useful properties do they share despite the Johnson solids not being vertex-transitive?

Schedule program demonstration

At the end, you briefly demonstrated the middle-school schedule program you built for a Cardputer M5.

Animated polar graph

You built and refined an animated polar curve in Desmos, adding controls for frequency, amplitude, diameter or displacement, a moving point and radius line, a faint preview of the complete curve, and a reference circle.

• Polar coordinates: The radius r determines distance from the center, while θ determines the angle. Restricting the θ-domain controls how much of the curve is drawn.
• Rose curves: For sine-based polar roses, odd frequencies produce the same number of petals as the frequency, while even frequencies produce twice as many. Over a full turn, odd-frequency curves retrace petals during the second half.
• Animation: A changing parameter was used to reveal the curve gradually, making it easier to see how the oscillating radius creates each petal.

Patterns in pi and Fibonacci numbers

You examined the famous run of six consecutive 9s beginning around the 762nd decimal place of pi and discussed why a truncation there can look misleadingly close to a terminating value. You also extended the Fibonacci sequence from 0 through 987, noticed convenient addition patterns, connected ratios of consecutive terms with φ, and interpreted 22/7 as 3 + 1/7, a close approximation to pi.

• Fibonacci sequence reviewed: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987.

Fractions, decimals, and divisibility

You practiced deriving decimal equivalents from division and related fractions instead of treating them as isolated facts.

• Place value: 0.1846 = 1846/10,000 = 923/5,000.
• Unit fractions: Examples included 1/6 = 0.1666…, 1/7 = 0.142857…, 1/8 = 0.125, 1/9 = 0.1111…, and 1/12 as half of 1/6.
• Divisibility: A number is divisible by 3 or 9 when its digit sum is divisible by 3 or 9. Divisibility by 6 requires being both even and divisible by 3.

Facts to learn through understanding

Review the Fibonacci sequence through 987 and learn the decimal equivalents of unit fractions through 1/12. Use relationships—such as halving a known fraction or decomposing division—to understand why each result is true.

Seeing a cube as parts

A cube can be expanded geometrically rather than through routine multiplication. A centered expansion has one core, six faces, twelve edges, and eight corners.

• One-sided form: (a+b)³ = a³ + 3a²b + 3ab² + b³.
• Centered form: (a+2b)³ = a³ + 6a²b + 12ab² + 8b³. This made 12³, 102³ = 1,061,208, and 104³ = 1,124,864 visible as sums of geometric pieces.
• Decimals: For 10.2³, use a core of 10 and layers of 0.1 on opposite sides, giving 1,061.208.
• Subtraction: Negative layers produce alternating signs; for example, 9³ = 10³ − 3·10² + 3·10 − 1 = 729.
• Approximation: Ignoring the thinner edge and corner pieces leaves the dominant face contribution, explaining why a small change in a³ is approximately 3a² times the change in a.

Higher-dimensional geometry

The same arithmetic pattern encodes the pieces of higher-dimensional cubes. A 4D hypercube has 1 hypervolume, 8 cubic cells, 24 square faces, 32 edges, and 16 vertices; orthogonal projection was understood as flattening a shape by removing a coordinate, like a distant-light shadow.

• Schläfli symbols: Examples included the cube {4,3}, octahedron {3,4}, tetrahedron {3,3}, icosahedron {3,5}, and dodecahedron {5,3}. Reversing the entries identifies dual pairs.
• Tessellations: {4,4}, {3,6}, and {6,3} describe regular planar tilings rather than closed solids.
• Open question: Which regular three-dimensional honeycombs can fill space indefinitely besides the cubic honeycomb?

Practice for the week

Do 20 minutes of mathematics every day, even when there is no school homework. Practice mentally cubing numbers by visualizing the core, faces, edges, and corners; include examples near round bases, decimals, and numbers slightly below a round base.

• Checkpoint: Reproduce 102³ = 1,061,208 using 100³, six faces, twelve edges, and eight corners.

Learning stance

Approach advanced ideas with curiosity rather than fear. The “autotelic” goal is to choose the next challenge that is neither too easy nor too large, then keep advancing one suitable step at a time.

Why the generations were updating incorrectly

In Java/Processing, assigning `game = temp` does not copy the array’s contents: both variables then refer to the same array in memory. This caused cells to be changed while the next generation was still being calculated. Copying each cell with nested loops kept the current and next generations separate.

Rules and wraparound boundaries

The implementation used Conway’s standard rules: an off cell turns on with exactly three live neighbors, while an on cell remains on with two or three neighbors. Adding the missing `case 0` fixed the remaining incorrect survivors. Boundary indices were wrapped with modulus, adding `N` before applying `% N` so negative indices correctly return on the opposite side of the grid.

Debugging and cleanup

The session tested neighbor counts, inspected individual cells, corrected grid ranges, and removed a `println` inside the update loop that was severely slowing the simulation. Repeated switch cases were then combined, with `default` handling the remaining outcomes more cleanly.

Clean up and practice the Game of Life code

Go through the working sketch and rewrite it more cleanly. Simplify or reorganize the code where useful, practice explaining how the rules and wraparound behavior work, and optionally adapt it so a user can create a custom starting matrix.

Game of Life structure

Represent each cell as 0 or 1 in a two-dimensional array. For every generation, count the eight surrounding cells while excluding the center: a non-living cell becomes alive with exactly three neighbors, while a living cell survives with two or three neighbors and otherwise becomes non-living.

• Two-grid approach: Read every cell from the current `game` array and write its next state to a separate `temp` array so updates do not affect later neighbor counts in the same generation.
• Neighbor coordinates: The lookup must use offsets from the current cell—`game[i+l][j+m]`—rather than checking the same corner repeatedly. Vlad identified and corrected this bug.
• Next debugging step: Every cell in `temp` must receive a value each generation. After the full generation is calculated, preserve two distinct arrays and swap them rather than making both variables refer to the same array.

Perlin noise exploration

Perlin noise produces smoothly varying, continuous randomness rather than independent static. Scaling its inputs changes the apparent clump size, adding time animates it, and using offset noise values for red, green, and blue creates flowing color patterns.

• Thresholding: Comparing a noise value with a threshold converts the smooth grayscale field into binary black-and-white cells.
• Terrain connection: Its smooth, clustered behavior makes Perlin noise useful for terrain, biome, and other natural-looking procedural generators.

Programming techniques practiced

The session used nested `for` loops, global constants, two-dimensional integer arrays, ternary conditional expressions, `switch` statements with required `break` clauses, casts between integers and floats, and debugging of null-pointer and out-of-bounds errors. Edge wrapping with modulus was attempted, but the temporary version instead skipped the outermost cells to get closer to a working program.

Continue the Game of Life

The implementation will be finished in the next session. The remaining work is to correct the next-generation array logic, assign every cell explicitly, decide how grid boundaries should behave, and verify that the simulation evolves correctly.

Prepare Processing

Get Processing working before the next session, using help from Vlad’s dad if needed, and paste the shared Game of Life code into Processing so it can be tested and continued.

Build a lasting project portfolio

Adam recommended maintaining a personal website that grows over time and contains projects such as the Game of Life, rather than making temporary standalone sites. He suggested obtaining a domain through Namecheap and using coding assistance only while continuing to understand and think through the code independently.

The golden ratio’s connected identities

The golden ratio is φ = (1 + √5)/2 ≈ 1.618. Its powers and reciprocals produce closely related values rather than unrelated decimals.

• Square: φ² = φ + 1 ≈ 2.618.
• Reciprocal: 1/φ = φ − 1 ≈ 0.618.
• Reciprocal square: 1/φ² = 2 − φ ≈ 0.382.
• Powers: φⁿ + φⁿ⁺¹ = φⁿ⁺², so consecutive powers follow the same addition pattern as Fibonacci numbers.
• Fibonacci connection: The ratio of successive Fibonacci terms approaches φ.

Deriving φ geometrically and algebraically

A segment was divided so that the ratio of the longer part to the shorter part equals the ratio of the whole segment to the longer part. Taking the longer part as 1 and the shorter part as x gives 1/x = 1 + x, hence x² + x − 1 = 0. Completing the square gives (x + 1/2)² = 5/4, so the positive solution is x = (√5 − 1)/2; therefore 1 + x = φ = (1 + √5)/2.

Area models and vocabulary

The area model for (a + b)² showed why the two cross terms cannot be omitted: (a + b)² = a² + 2ab + b². Desmos graphing and geometry tools were used to visualize the four regions, powers of φ, repeated line ratios, and the roots of a parabola. A radical is an expression containing a root; the quantity inside the radical symbol is the radicand.

Pico 2 W game demonstration

Vladimir showed a small Pico 2 W computer with a joystick and two buttons. It runs an offline brick-destroying game intended for road trips, with power-ups and a compact custom setup.

Possible direction for next session

Possible next projects are coding calculations or visualizations involving φ, or continuing Conway’s Game of Life, for which Vladimir has already begun setting up a matrix. The broader choice is whether to focus on pure coding or continue combining coding with new mathematics.

Orbits and escape-time coloring

Each iteration squares the current complex value and adds a fixed complex constant. Depending on the starting point and constant, the orbit may remain bounded, settle into a repeating cycle—such as the observed 7-cycle—or rapidly escape. A fractal image colors each pixel according to whether its orbit stays bounded and, if it escapes, how many iterations it takes to cross a chosen threshold.

Julia set versus Mandelbrot set

Both use the same recursive rule, but vary different inputs.

• Mandelbrot set: Start with z₀ = 0 and let each pixel represent a different value of c.
• Julia set: Hold c fixed and let each pixel represent a different starting value z₀; changing c produces a different Julia-set image.

Why shaders help

A CPU generally handles complicated work sequentially, while a GPU can perform many simpler calculations in parallel. Since every fractal pixel needs essentially the same independent iteration, GLSL/WebGL shaders can calculate thousands of pixels simultaneously.

Render a Julia set in Processing

Create a Julia-set image in Processing. Use a pixel array with loadPixels(), map screen coordinates to the complex plane, iterate z² + c for each pixel, and assign a color from the escape count. The teacher suggested that roughly 50–100 iterations and an escape threshold around 100 can be sufficient for an initial version.

• Supporting functions: Review map(), norm(), and lerp() for converting coordinates and color values between ranges.

The zeta function and infinite series

The session examined ζ(s) through sums of the form Σ(1/nˢ). An important distinction is that the ordinary series 1 + 2 + 3 + ⋯ diverges; −1/12 is the value assigned to ζ(−1) through analytic continuation, not its usual sum.

• ζ(1): 1 + 1/2 + 1/3 + ⋯ is the harmonic series, which diverges to infinity even though its terms shrink.
• ζ(2): 1 + 1/4 + 1/9 + 1/16 + ⋯ converges to π²/6.
• Continued values: ζ(0) = −1/2 and ζ(−1) = −1/12 are extended zeta-function values; the corresponding ordinary series do not converge.

Approximating π in Python

Vlad programmed the inverse-square sum and rearranged ζ(2) = π²/6 to approximate π. The working formula was π ≈ √(6Σₙ₌₁ᴺ 1/n²), with larger values of N producing a closer approximation.

• Key implementation detail: Start the loop at 1, accumulate 1/i², multiply the completed sum by 6, and take the square root only after the summation.
• Convergence experiment: Increasing the number of terms eventually produced approximately 3.14159, although this series converges slowly.
• Comparison: Changing 1/i² to 1/i demonstrated the harmonic series continuing to grow rather than approaching a finite value.

Derivatives and Newton’s method

For f(x) = x² − a, the derivative is f′(x) = 2x. The derivative gives the slope at each x-value: it is zero at the parabola’s center, negative on the left, positive on the right, and grows linearly.

• Vertical shifts: Moving a parabola up or down changes its roots but does not change the slope associated with each x-value.
• Newton update: Starting from an estimate c, the tangent line gives the next estimate c − f(c)/f′(c). Repeating this rapidly approaches a root.
• Square roots: Applying Newton’s method to x² − a = 0 approximates √a; the graphical construction approached √2 after only a few iterations.

Algebraic and transcendental numbers

The side discussion distinguished algebraic irrational numbers, such as √5 and the golden ratio, from transcendental numbers such as π and e. A transcendental number is not a root of any nonzero polynomial with rational coefficients.

Open threads for later sessions

Several ideas were introduced but intentionally left for deeper study.

• Zeta values: Why does analytic continuation give ζ(−1) = −1/12, and why does ζ(2) equal π²/6?
• Square-root program: Implement Newton’s method so the program calculates square roots without relying on a built-in square-root function.
• Modulo art: The teacher deferred an explanation of the one-equation modulo-art construction to a future lesson.