Gustav Freytag's 1863 Pyramid models a plot as tension T(t) rising from an inciting incident to a climax, then falling to resolution, across normalized story-time t ∈ [0,1]:
rising (t ≤ c): T(t) = B + (P − B) · s(t/c)^k s = smoothstep(x) = 3x²−2x³
falling (t > c): T(t) = R + (P − R) · exp(−5·(t−c)/(1−c))
complications: Ci(t) = Ai · exp(−((t − c·i/(N+1))² / (2σ²))), added into the rising branch
B = 0.15 baseline, R = 0.10 denouement floor, c = climax position, P = peak tension,
N = complication count, k = steepness, Ai grows with i (each raises the stakes further than the last)
This 2D-native companion runs the identical formula as the 3D ribbon (same B, R, smoothstep, Gaussian bumps, exponential decay — verified numerically against the 3D source) but reads it through four native plots instead of one oblique camera view:
- Tension arc (top) — the curve itself, filled to baseline, with pole flags at the five act boundaries and dots marking each complication bump — the same shape the 3D ribbon traces in space, drawn flat.
- Velocity trace — dT/dt plotted across the whole story at once. The 3D version only ever reports the reader marker's instantaneous velocity; here the full derivative curve is visible in one glance, including the sharp negative spike right after the climax that a single number can't show.
- Act timeline — the five acts (Exposition, Rising Action, Climax, Falling Action, Resolution) as a segmented bar with the current one highlighted, giving a compressed sense of story pacing independent of tension height.
- Phase portrait — tension on one axis, velocity on the other. As the reader marker travels the story, this traces a closed loop: a slow climb, a sharp spike near the climax, then a fast return leg during the fall. Shape and area of the loop change visibly with steepness and complications, something no spatial 3D view shows directly.
- Climax position, conflict intensity, complications — same meaning as the 3D version: c, P and N in the formula above.
- Rising-action steepness — exposes k directly as a slider (the 3D version only changes it via the Tragedy/Thriller presets); low k gives a slow, even climb, high k holds tension flat near baseline then snaps up right before the climax.
Real-world relevance: the same structural model is taught for Shakespearean tragedy, the five-act play, and — loosely — most modern thriller and screenplay outlines; changing c, P, N and k traces the difference between a slow-burn tragedy and a fast, complication-heavy thriller.