What a Slenderness Ratio Measures

Slenderness, the proportion Michael Stern’s supertall work starts from, is a ratio rather than a measurement of height. The Skyscraper Museum puts it plainly: it is the base width divided into the height, and structural engineers generally deem a tower slender once that ratio reaches 1 to 10 or 1 to 12, the proportion of an upended ruler.¹ Height and slenderness are separate quantities; a stout giant is not slender, and a narrow midrise can be. The entry traces how far the accepted ratio has moved, decade by decade.

Before 1916: shafts on expensive lots

New York built narrow towers before anyone measured them this way. Designing Buildings, drawing on the museum’s material, records that by 1909 the Metropolitan Life tower rose 700 feet and 50 storeys on a lot only 75 by 85 feet, in a period when no municipal rule constrained height and the value of the land plus the elevator and the steel cage made a city of towers.² The 1920s produced a second generation, pyramidal bases carrying thin shafts over a fourth of the lot or less, under the massing formula of the city’s zoning.² The history of those rules, and of the 1961 resolution that replaced them with a floor area formula, FAR, belongs to michaelsternjds.com, where that history is told; this entry needs only one consequence, that a developer could thereafter arrange the floor area a lot was allowed in any massing, and acquire a neighbour’s unused allowance.²

Figure

Tall against slender

	World Trade Center north tower	432 Park Avenue
Height	1,368 feet	1,396 feet
Base	209 feet on each side	93 feet square
Ratio	Under 1 to 7	1 to 15
Status when charted	World's tallest on completion in 1971	Under construction in May 2016

1961 to 2007: a ceiling of seven to one

For the decades that followed, the ratio hardly moved. Informed Infrastructure records that for generations of structural engineers a height to base ratio of 7 to 1 was thought to be the limit at which a building stayed rigid enough to keep wind acceleration below the point of unease or motion sickness.⁴ The museum’s own comparison makes the same point from the other direction: the World Trade Center north tower, the world’s tallest when finished in 1971, had a square plan 209 feet across and a ratio under 1 to 7.¹ Height was pursued; narrowness was not. Towers were offices, and offices wanted big floors, deep enough for rows of desks and a core that did not crowd them. The ratio stayed where the floor plate put it.

2007 onward: ten to one becomes the floor

The change came with residential towers on residential blocks. Informed Infrastructure dates the sprouting of slender buildings in New York to about 2007, driven by demand for apartments with views and by narrow parcels, and notes that zoning lets unused development rights be piled onto one lot, so tall heights on small footprints followed.⁴ The museum charted eighteen slender New York towers in 2016, starting at 1 to 10, and showed a more than five fold range in floor area among them, with the two smallest below 200,000 sq ft in area.¹ The Wikipedia entry’s list of examples carries the idea abroad and back: Highcliff in Hong Kong at 20 to 1 in 2003, 220 Central Park South at 18 to 1, Collins House in Melbourne at about 16 to 1, 432 Park Avenue at 15 to 1, One Madison Park at 12 to 1.³ The same entry notes that in structural engineering the word has an older, stricter meaning, a column’s effective length divided by its radius of gyration, which predicts buckling; the architectural ratio borrows the word without the mathematics.³

The building that made the new ratio legible, and that the rest of this cluster returns to, was 432 Park Avenue. Its structural engineer Silvian Marcus, writing in Structure magazine, gives the dimensions: 93.5 feet on each side, 1,396 feet high, 1 to 15, with 86 residential floors.⁵ Reaching that proportion took a spine and perimeter frame linked at five levels, upper slabs poured eight inches thicker for weight, five levels left open so the wind could pass, and a pair of 660 ton pendulums to keep accelerations inside the industry’s comfort band, all of it settled through repeated rounds with the wind tunnel laboratory.⁵

Figure

How one tower at 15 to 1 was made to stand still enough

1. 01
   A rigid spineA poured concrete spine around the vertical circulation, tied to an exposed perimeter frame, one tube inside another.
2. 02
   OutriggersOutriggers two storeys deep at five levels, linking spine to perimeter roughly where the open levels sit.
3. 03
   Mass where it countsFloor plates poured eight inches thicker on the upper storeys to add weight and limit displacement.
4. 04
   Open floorsFive double height levels left unenclosed, which weakened the vortices and the loads they produce.
5. 05
   DampersA pair of 660 ton pendulums near the crown, opposed to each other, holding accelerations within the comfort band the industry uses.

What the ratio costs

Carol Willis, who founded the museum, said in Architectural Record that New York’s narrowest towers need a really rigid concrete structure because they are subject to a tuning fork vibration from the wind, and that the weight of the concrete is what pulls the forces down to the ground.⁶ That weight is the ratio’s price. A 2021 thesis at MIT measured it: towers above 15 to 1 tend to use the most structure for each unit of floor, lower porosity means more material at any given ratio, and 432 Park Avenue’s structure carried roughly 1,024 kilograms of carbon dioxide equivalent per square meter against 460 for the runner up in the comparison set.⁷ Practical Engineering says why the material is spent on something other than strength: past about 10 to 1 the design is governed by the serviceability limit state, the movement that cracks tiles, breaks windows and unsettles residents long before anything fails, and the stiffening that would stop it eats the saleable area the whole venture depends on. The ratio also decides the elevator count, the core, the foundation and the construction method, each of which has its own entry elsewhere in this encyclopedia.⁸ The ratio, in other words, is a bill as much as a boast: it is paid in concrete, in carbon and in the comfort criterion that the next entry explains.