Gameplay

The product First Principles is named in part after Elon Musk’s comments on a first-principles approach to business and problem-solving in life and work; see First principles thinking & business for how that lines up with the graph and derivative gameplay.

Modes

  1. Graphing calculator — Functions are plotted on the Cartesian plane; numeric derivatives can be shown as a second curve (FunctionPlotter, LineRendererUI, DerivRendererUI).
  2. Derivative platformer — On Game scene, a small character runs on platforms generated from the curve; gaps / hazards follow rules based on the derivative (GraphObstacleGenerator, PlayerControllerUI2D).
  3. Graphing calculator mode — From the main menu only (green Graphing calculator button, right slot). Loads Game with no platformer; Back returns to Menu: a typed equation field graphs f(u) with variable x in the box (inner (u = k(x-D)); on-screen (y = A\cdot f(u)+C)). Trans (transformations: cycle (A), (k), (C), (D); double-tap nudges +, hold −) and Scale (zoom in on tap, zoom out on hold). Two-finger pinch on the graph changes the visible (x)-window. Supported syntax includes + − * / ^, parentheses, sin cos tan, asin acos atan, sinh cosh tanh, sqrt abs, log (base 10), ln, exp, min(,) max(,), pi, e, and implicit multiply (e.g. 2x, 2sin(x)).

Level parameters (colors, function type, transformation coefficients, story text) are defined in code via LevelManager sample levels / LevelDefinition (except this free-graph mode, which uses a fixed starter parabola). In platformer levels, FunctionPlotter auto-fits the vertical window to (f(x)) (and drag-polar traces when present) so flat curves fill the grid; (f’) is drawn with the same mapping. derivativeSafeThreshold still uses raw numeric (f’). Graphing calculator mode turns auto-fit off so Trans / Scale / pinch control the window.

Sample level lineup (GameLevelCatalog)

Titles are built at runtime from GameLevelCatalog.DisplayNames (order matches LevelManager sample levels). Expect 40+ named stages spanning primer → integrals → engineering → AP Calculus BC motifs → polar plots → Physics C hooks → aerospace teaching curves → two qualitative economics “index chart” stages (dot-com bubble mood; 2008 crisis mood — not real S&P data) → a final Mandelbrot escape-slice boss, including:

Block Examples
Primer + classics Primer, parabola, sine, cosine, absolute
Series / multivar Maclaurin e^x & sin, geometric tail, saddle & paraboloid slices
Integrals Area under the curve, Riemann left / right / midpoint
Engineering Damped oscillation, catenary (cosh), full-wave rectified sine |sin(x)|
AP BC + polar Arctan, logistic, Maclaurin cos, ln & √x, tan window, e^{kx}, cubic sketching, b^x; polar cardioid & rose; circle (upper semicircle / implicit form)
Physics C Exponential decay (τ / RC), projectile parabola, angular momentum / L = Iω (SHM framing)
Aerospace Lift (C_L(\alpha)) + stall, drag polar (three curves: parasitic, induced, total), isothermal (\rho(h)), phugoid/damped mode, Newtonian (\sin^{2}\alpha), Strouhal sine, re-entry decay envelope

Open Math tips & snippets on the level-select screen for short article text (including differentiation rules and definite vs indefinite integrals — game-style) plus blocks for first principles thinking (business) — see First principles thinking & business — and AMC 10/12, competition math, TMUA, MAT, AP Calculus BC, and AP Physics C. Longer notes: docs/math-concepts.md, docs/derivative-rules.md, docs/definite-indefinite-integrals.md, docs/engineering-math.md, docs/amc-10-12.md, docs/competition-math.md, docs/tmua-calculus.md, docs/mat-calculus.md, docs/ap-calculus-bc.md, docs/ap-physics-c.md.

Special levels can tint the grid and adjust how long story text stays on screen.

Controls (platformer)

Input Action
← / → or A / D Move
Space, W, or ↑ Jump
Gamepad (left stick + south face button) Supported via Input System
Touch (phones / tablets) On-screen ◀ ▶ and Jump when the build is mobile/touch-capable; safe-area layout keeps them above the home indicator. Keyboard still works on devices that have one.

Mobile-friendly UI

  • CanvasScaler uses reference 1080×1920 with a dynamic matchWidthOrHeight (via CanvasSafeAreaBootstrap): phones stay near 0.42–0.45; ~4:3 iPad / tablet sizes lean toward 0.5–0.52 so landscape and portrait both stay balanced.
  • Tablet detection uses shortest side in dp (DeviceLayout.IsTabletLike, threshold ~592dp) so high-res phones are not mistaken for iPads.
  • CanvasSafeAreaBootstrap wraps existing Canvas children in a _SafeContent root inset with Screen.safeArea (notches / rounded corners).
  • Runtime HUD (story, stage, controls hint, level-select list, math overlay) parents to that safe root where applicable.

Legacy Trans / Scale tuning buttons on the Game UI are disabled; tuning is driven by level definitions, not free sliders.

Stages

  • The run is split into stages (derivative “pop” moments at X thresholds).
  • A stage HUD (top-left) shows Stage k / n using the same typography family as the equation label.
  • Crossing a stage line triggers DerivativePopAnimator (visual emphasis on the derivative).

Level flow

  1. Menu — Entry and scene fade; footer shows Credits (ORCH AEROSPACE × GAME GENESIS), version, proprietary / initiative line, and Unity trademark notice (CREDITS.md for full attribution).
  2. LevelSelect — LevelSelectController builds a scrollable list from GameLevelCatalog.DisplayNames, plus Math tips & snippets (MathArticlesOverlay / LearningArticleLibrary). Choosing a level calls LevelSelection.SetSelectedLevel and loads Game.
  3. Game — LevelManager reads LevelSelection.ConsumeSelectedLevel, applies the level theme to FunctionPlotter, regenerates obstacles, and spawns / resets the player. The Menu scene shows title/version + credits (MenuCreditsBlock / SceneCreditsFooter.BuildMenuFooterRichText() → menu.version_line + menu.credits_line); gameplay HUD does not duplicate it above the controls hint. Per-level story banners use story.N from Localization/LevelStories/{locale}.txt when present (see Localization/README.md).

Spawn position

The generator picks a spawn column among the “safe start” columns where the platform top is lowest, so the player does not always appear at the visually highest part of the early curve.

Win / death

  • Hazards or falling below a Y threshold causes respawn on the same level.
  • Reaching the finish zone (far right) advances to the next sample level (wraps when last is complete).

UI polish

  • Story text appears at the top with the level name and fades (TMP).
  • Math concepts (top-right during play) opens the same scrollable LearningArticleLibrary reader as Level select’s Math tips & snippets — including how the main graph, derivative, platforms, Riemann strips, and stages map to calculus, plus the differentiation rules skill-tree block.
  • Controls hint bar at the bottom shows move / jump hints.
  • Stage panel uses a dark “glass” backing and accent strip consistent with the Limbo-like aesthetic.