May 25, 2010

Splitting a Trapezoid

In Washington, D.C., the National Gallery of Art's East Building, which opened to the public in 1978, features a façade that teases the eye. Designed by architect I.M. Pei, the massive structure is a regimented assemblage of vast walls, skewed polygons, and sharp edges. Walls unexpectedly meet at acute or obtuse angles instead of commonplace right angles.

To someone used to the relentlessly omnipresent right angles of more typical structures, viewing the East Building can be disconcerting. The relationships among angle, position, and perspective, born of long experience, no longer apply. One has to learn afresh how to view the building.

In commenting on the inspiration for his East Building design, Pei recalled, "I sketched a trapezoid on the back of an envelope. I drew a diagonal line across the trapezoid and produced two triangles. That was the beginning."

The trapezoid itself arose from the geometric shape of the building site, a plot bounded by four streets, with one running diagonally. Pei's design turned the trapezoid into an isosceles triangle and a smaller right triangle, a result possible because the wide side of the trapezoid was precisely twice the length of the opposite, parallel side.

Dividing the isosceles triangle into two right triangles adds to the unaccustomed abundance of acute and obtuse corners in the resulting structure. Indeed, serving as the structure's basic motif, triangles abound throughout the building.


Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photos by I. Peterson

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