Here is the toy. A gallery plan, six rooms, doors lined up on two axes like an enfilade. Every floor cell glows from dull to lime, and the brightest wears a lime star: the integration core, the spot from which the rest of the building is fewest visual steps away. Drag one partition across the middle doorway. The star jumps a room, and a number called intelligibility r² drops. You have asked a plan a question and got a number back: from where I stand, how much of this building can I already read?

The sketch runs on a 44 × 28 grid of 0.5 m cells. Every open cell is a node, and every pair of cells that see each other is an edge. The default gallery holds 1,028 cells, 257.0 m², and a full rebuild and analysis takes 25–107 ms. That is play speed, not render speed.

The rules

• ✎ wall paints walls and the analysis recomputes while you drag. ⌫ open erases them.
• ◎ probe shows any cell’s isovist in cells, m² and per cent of the plan.
• ★ is the most integrated cell. ◇ is the deepest.
• ⤓ grid CSV exports x_m, y_m, connectivity, mean_depth, integration per open cell.

Three papers, one grid

Michael Benedikt’s isovist (1979) is, in the words of the UCL-run spacesyntax.online glossary, the set of all points visible from a given vantage point. That is the grey wash under your probe. In 2001, Alasdair Turner, Maria Doxa, David O’Sullivan and Alan Penn at UCL published “From isovists to visibility graphs” in Environment and Planning B, turning every isovist into graph edges. Once the plan is a graph, you can measure it, using Bill Hillier and Julienne Hanson’s integration from The Social Logic of Space (1984), developed at the Bartlett, UCL, in the 1970s–80s:

1. Mean depth. Breadth-first search from every cell, averaging the steps to everything else.
2. Relative asymmetry (RA). Normalise that depth between two extremes: every space directly connected, or a single chain.
3. RRA. Divide RA by the D-value, the RA of the root of a diamond-shaped graph, where most spaces sit at middle depth and fewer sit above and below. Systems of different sizes become comparable.
4. Integration. Invert RRA, so higher means more integrated.

Packing that stack behind a mouse drag, in a browser, in a tenth of a second, is engineering too. The lineage only became a toy because someone shipped it.

The experiment that is the story

Switch the preset to cells · offset doors. Same rooms, with only the doors moved off-axis. Mean integration falls from 11.67 to 8.58. Intelligibility, Hillier’s test of whether a part can predict the whole, falls from r² 0.98 to 0.76. In the enfilade, what you see tells you how the plan connects. In the offset cells it no longer does. Visitors feel that before they can name it, and here it is a number.

←TODAY: a 28 KB browser sketch turns a plan into a visibility graph in about 100 ms. →3012: the Sentinels of Zurich-3012 read a plan’s legibility before it has a wall in it. Fulcrum: intelligibility is a design quantity you can fix in the drawing, long before signage has to fix it in the building.

Now break it

Open the blank plan. Every cell sees all 1,091 others, so RA is 0 and integration is ∞. That is not a bug; it is what the mathematics says about a room with no depth. Add one column. Wall off a corner and watch the warning count the cells cut off from the core. The lesson is in the misuse.

Where it earned its keep

Space Syntax Ltd worked on Trafalgar Square with Foster and Partners for the City of Westminster from 1996, and the first masterplan element was completed in 2003. The firm reports that pedestrian movement rose thirteen-fold afterwards, which is its own figure, not an independent measurement. For real work, the reference tool is depthmapX, released by the UCL Space Group under GPLv3 and kept alive by the contributors behind its 1,765 commits. It is invisible infrastructure that much plan analysis quietly relies on.

The Swiss anchor is a research group. Christoph Hölscher’s Chair of Cognitive Science at ETH Zürich studies navigation in complex public buildings. His 2012 study with Martin Brösamle and Georg Vrachliotis found that inexperienced visitors mostly used a central-point strategy, and argued that vertical connections must be modelled properly in visibility graph analysis. That is exactly where this sketch stops: one storey, eye level, a 0.5 m grid, a teaching model. Glass reads as solid wall, so the number is only as honest as the plan you feed it.

Atelier: Museum, school and hospital teams usually sign off on circulation by eye, then patch it with signage after handover. Integration and intelligibility give the competition-stage plan two numbers to defend instead. Monday move: export the grid CSV for your current scheme and map integration as a colour field onto the Rhino plan in Grasshopper, before any furniture goes in.

Hack: Compute Hillier and Hanson’s integration for one cell from its total depth. Feed it the total depth (TD) your breadth-first search returned and the system size n. Check: the end of a five-cell corridor has TD = 10, so integration(10, 5) returns 0.352, matching the hand-computed fixture.

from math import log2
def integration(TD, n):
    MD = TD/(n-1); RA = 2*(MD-1)/(n-2)
    D = 2*(n*(log2((n+2)/3)-1)+1)/((n-1)*(n-2))
    return float('inf') if RA == 0 else D/RA

Tonight, draw your own plan into the blank grid and write down two numbers: where its core sits, and how deep its deepest corner is.