The Verification Venue · perspective is constructed, not invented
The Floor That Builds Itself
The story goes that Brunelleschi invented perspective around 1415, a stroke of artistic genius — as if depth on a flat panel were a trick. It isn't a trick, and it isn't a style. Fix your eye at a distance, and the spacing of a receding tiled floor is not chosen. It is forced — there is exactly one correct answer, and the geometry hands you a built-in proof that you found it.
Below is a square-tiled floor. You aren't drawing it; you're setting an eye — how far you stand from the picture (the distance d) and where you look (the vanishing point). The floor then computes its own tile spacing. Drag the slider, or drag the vanishing point dot right on the floor. Watch the rows snap to the only positions they're allowed to take.
The self-building perspective floor
An interactive perspective floor. Use the distance slider, or drag the vanishing-point dot. Arrow keys move the vanishing point when the canvas is focused; up and down change the viewing distance.
How far the eye stands from the picture. Stand close and the floor plunges; stand far and it flattens.
Slide where the orthogonals (the lines running away from you) all meet. Everything obeys it.
Distance point — offset from centre
+2.00 tiles
always equals d — that's the proof
Diagonal closes on the distance point?
✓ yes — exact
residual < 0.001 tile
The pink line is the diagonal through the corners of a run of tiles. Extend it and it meets the horizon at the distance point (the yellow dot) — and the distance point sits at a horizontal offset from the vanishing point of exactly d, the viewing distance itself. That is the self-proof Renaissance workshops used: if your diagonals all land on one point, your tile spacing was right. Drag the distance and watch the dot track it, tile for tile.
Why is the spacing forced? A tile row at depth Z projects, by the single ray from the eye to the picture plane, to a drop below the horizon of p = h·d / (d + n·t) — eye-height h, distance d, tile depth t, row number n. Those drops are not equally spaced and not in a tidy ratio; they thin out as 1/(d+n·t). There is nothing to choose.
The check — the floor's numbers, recomputed live
For your current eye (d=2.0, h=1.0, t=1.0), the drop of each transversal below the horizon, straight from p = h·d/(d+n·t) — and the gap to the next row, to show it is neither a fixed step nor a fixed ratio:
| row n | depth d+n·t | p = h·d/(d+n·t) | gap to next | gap ratio |
|---|
The distance-point self-proof, with your current numbers:
Run it offline: node research/the-floor-that-builds-itself/verify-the-floor-that-builds-itself.mjs re-derives the drop law from the ray/plane intersection, confirms the diagonal vanishes at offset exactly d two independent ways, and proves the spacing is neither arithmetic nor geometric.
Verify a master
If perspective is forced geometry, the masters had to obey it — and you can check them. Below is a schematic reconstruction of the architecture in Leonardo's Last Supper (c. 1495–98): the coffered ceiling, the tapestry tops along each wall, the table edge. Every line you see running into the room is an orthogonal. Drag the two yellow probe lines so each lies along an architectural edge. The instrument reports where they cross — and it lands on Christ's head.
Find the vanishing point of the Last Supper
A schematic of the Last Supper's one-point architecture. Drag the endpoints of the two yellow probe lines onto the coffer or tapestry edges; the readout reports their intersection. Press R, or the reset button, to snap both probes onto two true architectural edges.
Your two probes cross at
— , —
drag the yellow handles onto the lines
Distance to Christ's head
—
documented VP region
The two probes you laid on the coffers and the tapestry meet within a small region at the centre of the wall, at the height of Christ's head — the documented vanishing point of the whole composition. The architecture isn't decoration around the figure; it's a perspective machine pointed at him. Leonardo set the one point of the construction on the one figure the painting is about.
The check — and an honest line about what this is
The mural is 880 × 460 cm, c. 1495–98, and its single one-point vanishing point is documented to lie at Christ's head / right temple, on the horizon, at the horizontal centre. This panel is a geometric reconstruction of that documented construction — not a pixel measurement of the actual (badly damaged, photographically distorted) painting. Because it is a genuine one-point construction, every orthogonal drawn here passes through the one point by definition; the check below confirms your two independently-placed probes recover it:
Reported as a region, not a magic pixel. "Converges near Christ's head" is a documented approximation: real published analyses of the mural place the vanishing point in a small zone by his right temple, not at a single certifiable pixel, because the surface is ruined and the photo is distorted. The verifier recomputes that all reconstructed orthogonals share one point and that the point is at the horizontal centre, above mid-height.
What's exactly true here, what's reconstructed, and what's history-by-hearsay
Exactly, provably true. The transversal-drop law p = h·d/(d+n·t) is the exact central projection of a flat floor onto a flat picture plane — re-derived in the verifier from the ray/plane intersection, not assumed. The distance point sits at horizontal offset exactly d from the vanishing point, proven two independent ways (the vanishing point of the 45° floor direction, and the extended line through two real projected tile corners) that agree to floating-point precision. The spacing is provably neither a constant step nor a constant ratio. Within the model, the spacing is not a choice — it is forced.
Idealised (named, not hidden). The construction assumes a flat picture plane, a single fixed eye (one cyclopean viewpoint, no binocular disparity, no head movement), and a perfectly square-tiled flat floor. Real paintings deviate: artists fudge spacing for composition, and many masters used empirical or synthetic perspective rather than the strict construction. The floor here is the ideal case the geometry describes; it is what reality is measured against, not a claim that every painting obeys it.
The Last Supper, honestly. The panel reconstructs the documented one-point geometry from the mural's published dimensions and the well-attested fact that its vanishing point lies at Christ's head. It is not a measurement of the painting's pixels: the mural began flaking within Leonardo's lifetime, has been repainted and restored repeatedly, and every photograph carries lens and viewing distortion — so the true vanishing point is a small region near his right temple, not a certifiable point. Treat the coincidence as "the documented construction recovers the documented point," which is exactly what it is.
History by hearsay. Brunelleschi left no written method. The famous c.1415 demonstration with a mirror and a painted panel of the Florence Baptistery is reconstructed from later accounts — chiefly Antonio Manetti's biography (c.1480s) and Vasari (1550) — and the date and details should be read as reconstruction. The first written codification of the construction is Alberti's De Pictura (1435); the "distance point" as an explicit checking device is later still (associated with the northern and French practitioners, formalised by the 16th–17th centuries). The mathematics on this page is timeless; the attributions around it are not certainties.