![]() Light caused the folding cube to cast a shadow, each of the four side squares would be replaced by a shadow rectangle, which would shrink back to a side of the original square. Object, it would be necessary to create unstraight dihedralĪngles, and all but the original square would simply disappearĪs the other squares folded into three-space. Of the individual pieces, but when it came time to assemble the AnĮngineer in Flatland could supervise the accurate construction ![]() ![]() Square, which we place at the bottom of the cross pattern. Yet attached to anything else, and these are ready for the last These squares are to be attached to which, giving the fold-out One of the square faces and attach the four adjacent squares to A preparatory crew in Flatland could begin the assembly, but they could not complete the project.Ī good example is the fold-out pattern for a cube. We can design a prefabricated polyhedral structure in the plane by laying out the polygons and indicating which edges are to be attached to which. We encounter similar challenges when we move from the plane into three-space. Overlapping construction of a collapsed square. We have to go to the plane to obtain a square, having all edges the same length and all angles equal. Of course these are the only kinds of angles available in Lineland. But he still could not construct a true square in the line since this "collapsed" quadrilateral has two different kinds of angles, two of them straight angles and two of them zero angles. If it were possible for two sides to occupy the same space on the line, the King could just place one hinged rod with two sides on top of a similar figure and connect their endpoints. He could not make the fourth connection without having the sides overlap. Times the length of the sides of the square. King could begin the assembly by making three of theĪttachments, but he would be left with just a hinged rod four Telling which endpoints should be attached to each other. Square, just four segments of equal length with instructions It wouldīe possible to give the King of Lineland a kit for building a To illustrate this device, we go back to lower dimensions. In order to gain a better appreciation of the global structure, the shape of the object as a whole, we can use the device of fold-out patterns. Most of our argument so far has been in terms of the local structure of a polyhedron or a polytope-that is, the faces in the neighborhood of a point or an edge. Fold-Out Patterns in Different Dimensions Fold-Out Patterns in Different Dimensions
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