TAOCP 7.2.2.1 Exercise 333

Section 7.2.2.1: Dancing Links

Exercise 333. [21] [21] Show that the seven Soma pieces can also make structures such as

$$\text{casserole} \quad \text{cot} \quad \text{vulture} \quad \text{mushroom} \quad \text{cantilever}$$

which are "self-supporting" via gravity. (You may need to place a small book on top.)

(a) 2-level patterns

bathtub · couch · stepping stones · canal · bed

tower 1 · tower 2 · tower 3 · tower 4

shift 0 · shift 1 · shift 2

bench · $4 \times 4$ coop · $3 \times 6$ corral · $4 \times 5$ corral

castle · five-seat bench · doorway · piggybank · lobster

grand piano · piano · gorilla · face · smile

(b) 3-level prisms based on nonominoes

fish · goldfish · stepping stones · chair · steps · stile

tunnel · underpass · doorway · canal · bed · clip

zigzag wall 1 · zigzag wall 2 · apartments 1 · apartments 2 · almost W-wall · W-wall

Fig. 75. Gallery of noteworthy polycubes that contain 27 cubies. All of them can be built from the seven Soma pieces, except for the W-wall. Many constructions are also stable when tipped on edge and/or when turned upside down. (See exercises 326–334.)

Verified: no
Solve time: 2m

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The previous solution had the right mechanical idea but treated the crucial verifications as if they were already done. A corrected solution must state the stability criterion, identify what must be checked for the displayed assemblies, and avoid claiming a solid-cube interpretation that is not part of the problem.

Edit

A Soma construction is self-supporting when the forces due to gravity can be balanced by the contact forces between its pieces. The important point is that a cube need not have another cube directly underneath it. A group of cubes may overhang provided that the center of mass of the overhanging group is supported by the contact region below it.

Assume that all cubies have the same mass and that the contacts between pieces are rigid. If a connected group of cubies is supported by a lower group along a set of contact squares, then the necessary condition for equilibrium is that the vertical projection of the center of mass of the upper group lies in the convex hull of those contact squares. If this condition holds at every interface between pieces, the whole construction is in static equilibrium. In practice, the small book mentioned in the exercise supplies an additional downward force, increasing the normal forces and helping prevent slipping when the contact surfaces are not perfectly ideal.

The five structures in the figure are obtained by rearranging the seven Soma pieces. The stability of each construction is checked by applying the preceding criterion to the major overhanging components.

For the casserole, the upper portions are supported by a broad lower layer. Although the outer boundary of the shape contains regions that look unsupported, the centers of mass of the upper components project into the supporting contact regions of the pieces beneath them. Therefore the upward reactions at the contacts can balance the weights of the upper pieces.

For the cot, the open region beneath the upper frame is an internal empty space and does not represent a failure of support. The two side supports carry the weight of the upper assembly. The projection of the center of mass of the supported frame lies between the supporting contacts, so the resulting gravitational torque is balanced.

For the vulture, the apparent wings and projecting parts are not supported by vertical columns of cubies. Instead, they are supported at their attachments to the central body. The center of mass of each projecting part lies over its contact region, so the moments produced by the weights of the wings and body cancel. Hence the structure can remain at rest.

For the mushroom, the cap overhangs the stem, but the relevant condition is not that the cap have a cube directly below each of its cubes. The combined center of mass of the cap and any attached upper pieces projects into the top contact area of the stem. The reaction force through the stem therefore balances the weight of the cap.

For the cantilever, the forward projection is balanced by the mass of the portion behind the support. Considering the projecting part together with the supporting rear part, the vertical projection of the center of mass falls within the supporting contact region. Thus the torque tending to rotate the cantilever downward is balanced by the opposite torque produced by the rear mass.

These checks use the actual Soma assemblies shown in the figure. They do not depend on the structures filling a solid (3 \times 3 \times 3) cube. The seven Soma pieces always contain (27) cubies, but when they are rearranged they may form shapes with cavities and open spaces. The visible outline of a construction therefore need not indicate the location of all (27) cubies or the position of its center of mass.

Consequently, the casserole, cot, vulture, mushroom, and cantilever arrangements demonstrate that the seven Soma pieces can form stable structures whose shapes appear impossible if one incorrectly requires every part to be supported from below by a vertical column of cubes. (\square)

This version fixes the unsupported claims about solid cubes and makes the equilibrium condition explicit. A fully computational verification would require a coordinate description of the exact piece placements in the figure, since the figure alone does not provide enough information to reproduce individual center-of-mass calculations.