Painting a Pyramid.Pdf

Delving deeper christopher M. Kribs-Zaleta Painting the Pyramid ne hallmark of a mathematician is the can measure or count about a cube: numbers of ver- instinct to extend and generalize. This article tices, edges, and faces, surface area, and volume. In Otakes a common high school problem relating particular, the unit cubes at the vertices of the large algebra, geometry, and patterns and extends it. The cube are those that have paint on 3 faces (for n > 1). extension seems simple enough at first glance but The remaining unit cubes along the edges of the large proves to have a number of interesting complications. cube are those that have paint on 2 faces, and the cor- Painting the Cube (NCTM 1989; Reys 1988) is a responding formula 12(n – 2) that emerges from the popular and mathematically rich problem used in mid- data is formed by measuring the length of one edge, dle and high school mathematics courses (as well as discounting the two corner cubes, and then multiply- some courses for preservice teachers). In that problem, ing by the number of edges. The remaining unit a large cube is assembled from small unit cubes, and cubes on the faces of the large cube are those with the exposed faces of the large cube are then painted one painted face, and they are counted by measuring (see fig. 1). The problem asks how many of the unit the area of a face, discounting the edges and corners, cubes will have paint on 0, 1, 2, . faces, as a func- and multiplying by the number of faces. Finally, the tion of the edge length n of the large cube (measured number of completely unpainted cubes inside the in small cubes). Painting the Cube blends algebra and large cube, as well as the total number of unit cubes geometry in a way that allows students at various needed to form the large cube, is given by invoking levels to use the concrete context to explore linear, the volume formula for a cube. quadratic, and cubic functions and relationships. High Now suppose we change the problem slightly, to school and college students typically approach this consider a “pyramid” made in the same way, namely, problem by building models for the first several cases, a tetrahedron of side length n formed with unit tet- gathering data in a table, generalizing the patterns into rahedra. The analogous question appears to be, If we formulae, and finally justifying the formulae. assemble a number of small unit tetrahedra to form The formulae that arise in this problem have a a large tetrahedron with edge length n, how many very nice correspondence to all the quantities one small tetrahedra will have paint on 0, 1, 2, 3, or all 4 faces? The alert reader, recalling that tetrahedra do this department focuses on mathematics content that appeals to secondary school not tessellate three-dimensional space, may suspect teachers. it provides a forum that allows classroom teachers to share their mathemat- that the question is not quite as simple as it sounds. ics from their work with students, their classroom investigations and projects, and their other experiences. We encourage submissions that pose and solve a novel or UnDeRSTAnDing THe QUeSTiOn interesting mathematics problem, expand on connections among different math- If we follow the basic approach described above for ematical topics, present a general method for describing a mathematical notion Painting the Cube, the first thing we will do is con- or solving a class of problems, elaborate on new insights into familiar secondary struct models for small n, gather data, and look for school mathematics, or leave the reader with a mathematical idea to expand. send submissions to “Delving Deeper” by accessing mt.msubmit.net. patterns. (Go on; get yourself a bunch of little tet- “Delving Deeper” can accept manuscripts in ASCII or Word formats only. rahedra and play along.) The case n = 1 obviously corresponds to a single unit tetrahedron with paint Edited by Al Cuoco, [email protected] on all 4 faces. When we attempt to assemble the n = Center for Mathematics Education, Education Development Center 2 case, however, we run into trouble—as the earlier Newton, MA 02458 remark about tessellating three-space should have prepared us to expect. As shown in figure 2, when E. Paul Goldenberg, [email protected] Center for Mathematics Education, Education Development Center we place four unit tetrahedra toe-to-toe to cover the Newton, MA 02458 vertices of the larger tetrahedron, we see that the space left in between them is not tetrahedral. If we 276 MatheMatics teacher | Vol. 100, No. 4 • November 2006 Copyright © 2006 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. Number of unit cubes with paint on… edge length 0 faces 1 face 2 faces 3 faces TOTAL 1 (1) 2 8 8 3 1 6 12 8 27 4 8 24 24 8 64 n (n – 2)3 6(n – 2)2 12(n – 2) 8 n3 Fig. 1 a 3 × 3 × 3 cube and data gathered for the Painting the cube problem Three- dimensional Headprint Footprint Fig. 2 Four-unit tetrahedra stacked to make a 2 × 2 × 2 tetrahedron consider the vacant space in the middle as a solid, we see that four triangles touch each corner of this new solid: It is octahedral in shape. It is at this point that we realize we must rephrase the questions we are investigating. To get a complete understanding of the make-up of the large tetrahedron, we will want to know what kinds of objects are required to build the nth large tetrahedron, how many of each type are required, and how many of Fig. 3 three views of each of the top four layers of a large tetrahedron. White figures them have paint on how many faces. In fact, as we are upright unit tetrahedra, shaded figures are octahedra, and black figures are inverted will see shortly, there is one more type of object to unit tetrahedra. count, which does not appear until the third layer. shape, denoted in black in figure 3, is also required eXPlAining THe PATTeRnS to build subsequent layers of the larger tetrahedra, Figure 3 shows the nth layer of a multilayered tet- so we add a column to our data tables (see table 1), rahedron (counting down from the top) for n = 1, 2, as one more type of object to count. This is the last 3, 4. If we count and sum each of these (e.g., adding type of object required to build the larger tetrahe- the first four layers for the n = 4 case), we will gen- dra, although it is not until n = 5 that an unpainted erate the data in tables 1 and 2. upright tetrahedron appears (all the inverted tet- Note that when adding a layer (moving from the rahedra are unpainted), visible in the center of the n = 3 to n = 4 case, for example), the counting must footprint of the fourth layer in figure 3, and it is not be done anew. Some tetrahedra may have paint until n = 6 that an unpainted octahedron appears. on a different number of faces in the larger case, There are many patterns to explore in these because a given tetrahedron may no longer be on data, even after they have all been identified and an outside edge or face. the formulae derived. Veterans of Painting the Note also that in the three-dimensional view of Cube will think to observe that unit polyhedra with the third layer, it is evident that when we place three paint on 3 faces are again corner pieces, polyhedra octahedra among the six upright unit tetrahedra, with paint on 2 faces occur along edges, polyhedra there is a space left in the center of the layer, between with paint on 1 face appear in the centers of faces the octahedra. The shape of the space is that of an of the large tetrahedron, and polyhedra with paint inverted (i.e., point-down) unit tetrahedron. This on no faces fill the interior. Indeed, the formulae Vol. 100, No. 4• November 2006 | MatheMatics teacher 277 Table 1 Numbers of Unit Tetrahedra Used in Making Large Tetrahedra Number of upright tetrahedra with paint on . edge Inverted length 0 faces 1 face 2 faces 3 faces TOTAL (no paint) 1 (1) 2 4 4 ()n−4() n − 3 ()n − 2 3 6 4 10 1 6 4 4 12 4 20 4 5 1 12 18 4 ()n−435() n − 3 ()n − 2n() n +110() n + 2 6 6 6 4 24 24 4 56 20 ()n−4() n − 3 ()n − 2 n() n +1()n + 2 ()n−2() n − 1 n n 2(n – 3)(n – 2) 6(n – 2) 4 6 6 6 Table 2 n() n +1()n + 2 ()n−2() n − 1 n (n − 5)(n− 4)( n − 3) Numbers of Unit6 Octahedra Used in Making Large Tetrahedra 6 ()n−4() n6 − 3 ()n − 2 6 Number of octahedra with paint on . edge ()n−2() n − 1 n (n − 5)(n− 4)( n − 3()n) −1 n()n + 1 length 0 faces()n− 46 () n − 3 ()n − 2 1 face 2 faces 3 faces 6 n() nTOTAL+16()n + 2 1 6 6 (n − 5)(n− 4)( n − 3) ()n−1 n()n + 1 2 (1) n() n +16()n + 2 6 ()n−2() n − 1 n 3 4 4 6 6 4 ()n−1 n()n + 1 6 4 10 5 ()n−26 () n − 1 n 4 12 4 (n − 520)(n − 4)( n − 3) 6 6 6 1 12 18 4 35 (n − 5)(n− 4)( n − 3) ()n−1 n()n + 1 n 2(n – 4)(n – 3) 6(n – 3) 4 6 6 for numbers of()n upright−1 n()n +tetrahedra 1 parallel exactly edge, subtracting 2 to account for the corners, and the formulae for unit6 cubes in Painting the Cube.

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