CUBE-NETS AS A STUDENT MATH RESEARCH PROJECT Reginald Luke, Ph.D. with students, Mara Olivares and Sindhu Murthy Introduction and Background qRetired from community college (Middlesex CC) teaching, mainly, precalculus, calculus, and linear algebra, and adjuncting at Rutgers University in graduate statistics. qReflecting on student math research work as a few enterprising students had approached me to engage in the doing research beyond typical homework. qFirst, Mara Olivares, who got involved with cutting cubes and counting paths. She was interested in a NASA scholarship and later transferred to Montclair U majoring in math education. qLater, Sindhu Murthy, who focused on trees and symmetry, and presented her math results on a Science Day at MCC. She graduated MCC as valedictorian and transferred to Rutgers U. The Cube-Nets Puzzle from NCTM Illuminations website page qInteractive geometric puzzle for elementary and middle-school students (from https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Cube- Nets/ qParenthetically, the site states that there are 11 different (non-isometric)Cube- Net configurations, which we labelled C1 to C11. qThe question I posed to my students was: Why are there 11 such configurations? qThe first student researcher, Mara, began cutting edges of a cube and tracing paths which ultimately allowed an unfolding of the cube into a planar polyhedron, 6 contiguous squares, the Cube-Nets designated C1 to C11. The 11 Cube-Net Configurations, C1 to C11 Trees and Minimum Spanning Trees qWe realized that the cube cutting paths were spanning trees that covered all 8 vertices of the cube, since each corner of the cube had to be unzipped to become planar. No loops were allowed since the cube-nets were connected polyhedral. The spanning tree had to have nodes of degree 3 or less. qWe found 6 different minimum spanning trees, none with more than two deg 3 nodes and none with an even number of edges between deg 3 nodes. Surprisingly, trees with 8 vertices and 7 edges that did not make the cut include (right figure): Counting Minimum Spanning Trees Kirchoff’s Matrix-Tree Theorem (1847) Hypercube Version (Stanley, 1999) qLet G(V, E) be a graph and the qFor any hypercube, Qn (Q3 being the Laplacian matrix L = D – A, where D is 3-d cube), the determinant value from the degree matrix (with values for Kirchoff’s Theorem works out to be: each vertex along the main diagonal) and L the adjacency matrix with n 2 – n - 1 $ C(n,k) values 1 or 0, depending on whether 2 ∏!"# � there is an edge or not between the two vertices being considered (note qKirchoff is usually known for his the zeroes along the diagonal). Then connection to electrical circuits the number of spanning trees of G is and chemistry/spectroscopy. the determinant of any cofactor of L. A Simpler Case: The Tetrahedron (Smallest Platonic Solid) � −� −� −� L = D - A = −� � −� −� −� −� � −� −� −� −� � � � � � � � � � � −� −� D = � � � � A = � � � � One cofactor of L = −� � −� � � � � � � � � −� −� � � � � � � � � � with determinant value of 16. By Kirchoff’s Matrix-Tree Theorem there are 16 different ways to cut open a tetrahedron and unfold it into a planar figure, a Tetra-Net. How many are there of these? A Simpler Case: The Tetrahedron (Only two spanning sets leading to two Tetra-Nets) q Linear spanning set: How many ways qOne deg 3 vertex: There are 4 nodes to to linearly cut open a tetrahedron? position this branched spanning set. q There are 4 starting vertices possible, THUS, we have a total of 12 + 4 = 16 leading to 3 ending vertices, but each different spanning sets that can cut path is reversible. So there are 4 . 3/2 = open a tetrahedron into two possible 12 ways to linearly cut it open. configurations. ◦ The Cube-Net Case qLessons from the Tetra-Net case: The spanning tree cuttings will create the perimeter edges of the Tetra-Net (darker shaded), and leave the interior edges between contiguous triangles untouched. This will help the visualization in the case of Cube-Nets, to determine which square faces lie adjacent. qThe use of Kirchoff’s Theorem applied to the Cube leads to a 8 x 8 Laplacian matrix and requires the calculation of the determinant of a 7 x 7 cofactor. Think n 2 – n - 1 $ C(n,k) about that! We use the alternate Hypercube Formula: 2 ∏!"# � = 24 (13 )(23)(31) = 384 different Cube spanning trees. qThis result already appears in Tulley (2012) and uses a more sophisticated technique. For us, the question was how the 6 cube-spanning sets generating 384 different possibilities led to the 11 cube-net configurations, using only combinatorics and geometric visualization, a more simplistic approach, which we now continue. Cube- Nets: The Linear Spanning Tree, Case 1: End Nodes of Tree Cross Cube Diagonal qWe attempt to fit a linear spanning tree qThe tree diagram of possible spanning trees is shown between pairs of vertices lying across the below with the topmost branch for the diagram on left. cube’s main diagonal, as below, V1 - V8. Note that by our vertex labeling the parity of vertex Three other possible pairings are V2 – V5, numbering is alternatively odd and even. Thus, our V3 – V6 & V4 – V7. linear spanning set of 7 edges and starting at V1 cannot ever end at V3, V5 or V7. qThere are 6 different linear spanning trees (two junctions of 3 and 2 branches) possible for the specific vertex pair (V1 - V8). Since there are 4 such cross diagonal vertex pairs, we get a total 24 spanning trees for this Linear 1 case. Linear Tree 1, Cross Vertices, Cube Unfolded to C4 qSimilar to this linear V1-V8 example, all of the 24 linear L1 spanning trees, cuts open the cube to become Cube-Net C4. The cube faces are labeled to assist in the unfolding visualization. Cube- Nets: The Linear Case 2, Adjacent Vertices qAgain envision a 7-edges linear spanning tree starting at V1 and ending at an adjacent even-numbered vertex, V2, V4 & V6. We only show the graphics for V1 to V2 going first through V4. The results for traversing V6 first is similar from symmetry qThere are 4 odd and 4 even-numbered vertices, thus 16 vertex pairs. We already covered the 4 across-main diagonal one, leaving 12 for adjacent vertex pairs. Double this amount for whether we go through V4 or V6 first, so there should be 24 Cube-nets for each of the cube cuttings below. But what are they? the same? Cube Unfolding Using a Linear Spanning Tree Between Adjacent Vertices The Linear Spanning Tree L2, that connects adjacent vertices, surprisingly unfolds into two different Cube-Nets, C1 and C10. There are 24 of each configuration type. Spanning Trees with One 3-deg Node Fitting qSince this tree is not symmetric, place the degree 3 node at V1 (WLOG for any of 8 possible vertices) and fit the branches with 1, 2 and 4 edges emanating from V1 for 3! = 6 variations. Thus, depending on how the rest of the edges end up, we already have a multiplier of 48 for this spanning tree. In what follows, we find out there are 3 different ways to complete the 1, 2 & 4 edges branching emanating from V1 (WLOG) designated Types 1, 2 & 3. Spanning Trees with One 3-deg Node, Types 2 & 3, Fitting Type 2 Type 3 Spanning Tree with One 3-deg Node, with 2, 2 & 3 branch edges . qFit the deg-3 node to V1, WLOG 1 of 8 vertices, and direct the 3 edges branch along V4 (WLOG 1 of 3 directions). We find two different paths, . based on how the two 2-edge branches are placed. Both cuttings yield the same Cube-Net. Thus, 48 such spanning trees are associated withasaa with C 9. Spanning Trees with Two 3-deg Nodes Fitting ◦ This spanning tree is asymmetric, so place the 1-1-1 deg 3 node at V1 (1of 8) and node2 at V4 (WLOG 3 ways). Then the shorter 1 edge branch either goes to V3 or V5 (2 choices) and the 3-edge branch fits neatly in. Thus, this configuration has 8.3.2 = 48 different spanning trees, all leading to the C8 Cube-Net configuration. Only shown: (In other, switch 3 and 5.) Spanning Trees with Two 3-deg Nodes Fitting qA symmetric spanning tree, so place one 1-2-1 deg 3 node at V1 (1of 8) and the other at V4 (WLOG 3 ways). But this is reversible, so count 12 ways. Next at V1 set the 1-edge branch to V6 (or V2, so 2 ◦ c ways). The 2-edge branch at V2 must go to V7, not V3 Thus, 24 variations so far. But in completing the 2nd 1-2-1 node we find two possibilities, as shown on the right, leading to different Cube-Nets, given on the next slide, with 24 spanning trees each embeddable within the cube. Cube-Nets for Spanning Tree with Two 3-deg Nodes: Fitting Spanning Trees with Two 3-deg Nodes, Fitting the last one . q Place one node of this symmetric spanning tree at V1 and the longer branch towards V4 (1 of 3 choices). The other node must end up at V8, so there are 4 ways in the cube that these nodes can be paired. So 12 variations are possible. However, at ◦ . V4 the 3-edge branch can alternatively visit V3 or V5, given another factor of 2, for a total of 24 ways to embed the spanning tree. Surprisingly, both variations lead to Cube-Net configuration C 11. Cube-Stats: 6 Spanning Trees, 384 Variations, and 11 Cube-Nets Tree Type Variation # Cube-Net Diagram Linear: End Nodes Cross Diagonal 24 C 4 Linear: End Nodes Adjacent 24 C 1 Linear: End Nodes Adjacent 24 C 10 One 3-deg Node 1-2-4 branches 48 C 7 One 3-deg Node 1-2-4 branches 48 C 2 One 3-deg Node 1-2-4 branches 48 C 3 One 3-deg Node 2-2-3 branches 48 C 9 Two 3-deg Nodes 1-1-1-3-1 48 C 8 Two 3-deg Nodes 1-2-1-2-1 24 C 5 Two 3-deg Nodes 1-2-1-2-1 24 C 6 Two 3-deg Nodes 1-1-3-1-1 24 C 11 TOTAL 384 Relevant and Related Literature ◦ Turney (1984) forms trees from the Cube-Nets (vertex being a face and edge if two faces are contiguous) and shows how to associate the 11 Cube-Nets with ”paired” 6-node trees.