CUBE-NETS AS A STUDENT MATH

RESEARCH PROJECTReginald Luke, Ph.D. with students, Mara Olivares and Sindhu Murthy

Introduction and BackgroundqRetired 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 TreesqWe 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 TreesKirchoff’s Matrix-Tree Theorem (1847)

qLet G(V, E) be a graph and the Laplacian matrix L = D – A, where D is the degree matrix (with values for each vertex along the main diagonal) and L the adjacency matrix with values 1 or 0, depending on whether there is an edge or not between the two vertices being considered (note the zeroes along the diagonal). Then the number of spanning trees of G is the determinant of any cofactor of L.

Hypercube Version (Stanley, 1999)

qFor any hypercube, Qn (Q3 being the 3-d cube), the determinant value from Kirchoff’s Theorem works out to be:

22n

– n - 1 ∏!"#$ 𝑘 C(n,k)

qKirchoff is usually known for his connection to electrical circuits and chemistry/spectroscopy.

A Simpler Case: The Tetrahedron(Smallest Platonic Solid)

D =𝟑 𝟎𝟎 𝟑

𝟎 𝟎𝟎 𝟎

𝟎 𝟎𝟎 𝟎

𝟑 𝟎𝟎 𝟑

A = 𝟎 𝟏𝟏 𝟎

𝟏 𝟏𝟏 𝟏

𝟏 𝟏𝟏 𝟏

𝟎 𝟏𝟏 𝟎

L = D - A =𝟑 −𝟏−𝟏 𝟑

−𝟏 −𝟏−𝟏 −𝟏

−𝟏 −𝟏−𝟏 −𝟏

𝟑 −𝟏−𝟏 𝟑

One cofactor of L = 𝟑 −𝟏 −𝟏−𝟏 𝟑 −𝟏−𝟏 −𝟏 𝟑

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

to linearly cut open a tetrahedron?

q There are 4 starting vertices possible, leading to 3 ending vertices, but each path is reversible. So there are 4 . 3/2 = 12 ways to linearly cut it open.

◦

qOne deg 3 vertex: There are 4 nodes to position this branched spanning set. THUS, we have a total of 12 + 4 = 16 different spanning sets that can cut open a tetrahedron into two possible configurations.

The Cube-Net CaseqLessons 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 about that! We use the alternate Hypercube Formula: 22

n– n - 1 ∏!"#

$ 𝑘 C(n,k)

= 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 DiagonalqWe attempt to fit a linear spanning tree

between pairs of vertices lying across the cube’s main diagonal, as below, V1 - V8. Three other possible pairings are V2 – V5, V3 – V6 & V4 – V7.

qThe tree diagram of possible spanning trees is shown below with the topmost branch for the diagram on left. Note that by our vertex labeling the parity of vertex numbering is alternatively odd and even. Thus, our 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 C4qSimilar 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 VerticesqAgain 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 VerticesThe 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 NodeFitting

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 spanningtrees are associated withasaa

with C 9.

..

Spanning Trees with Two 3-deg NodesFitting

◦ 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, switch3 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 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.

◦ c

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 DiagramLinear: End Nodes Cross Diagonal 24 C 4Linear: End Nodes Adjacent 24 C 1Linear: End Nodes Adjacent 24 C 10

One 3-deg Node 1-2-4 branches 48 C 7One 3-deg Node 1-2-4 branches 48 C 2One 3-deg Node 1-2-4 branches 48 C 3One 3-deg Node 2-2-3 branches 48 C 9Two 3-deg Nodes 1-1-1-3-1 48 C 8Two 3-deg Nodes 1-2-1-2-1 24 C 5Two 3-deg Nodes 1-2-1-2-1 24 C 6Two 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. He then uses this technique to show that there are 261 unfoldings of the 4-dim Cube, the Tesseract. Jobbins (2015) follows up this technique “paired trees” using coloring and shows how to enumerate nets of various polyhedral. Tuffley (2012) gives a direct combinatorial proof that the 3-cube has 384 spanning trees, using an “edge slide” operation that is much more sophisticated then our approach. An edge slide graph of the 3-cube has vertices the spanning trees of the 3-cube and an edge if two trees related by a single edge slide. Henden (2011) was able to explore one 64-vertex component of this large graph as an undergraduate summer research project. Harary et al (1988) provides an extensive survey of the theory of Hypercube graphs for any budding graph theory researcher.

q Lessons Learnt

qEnterprising students at the community college level are able to conduct and benefit from math thinking beyond the typical homework variety, especially, research directed to answering puzzling questions. At this level, the experiences of constructing tangible models, developing geometric visualization skills and doing combinatoric calculations were invaluable.

qWe learnt to examine simpler cases, such as that of the Tetrahedron.qThe experience included delving into what other mathematicians had

previously produced in the literature and the diverse techniques utilized. In our case, we stuck with simple combinatorial counting, but surprisingly could not avoid concepts from discrete math and graph theory, such as spanning sets and Kirchoff’s Matrix-Tree Theorem.

References◦ Harary, Frank, Hayes, John P. & Wu. A Survey of the Theory of Hypercube Graphs. Computer

Mathematics Applications, Vol. 15 (4), pp 277-289.

◦ Henden, Lyndal. (2011). The Edge Slide Graph of the 3-Cube. Rose-Hulman Undergraduate Mathematics Journal, Vol. 12 (2), Fall 2011.

◦ NCTM Illuminations. Retrieved from http://www.nctm.org/Classroom-Resources/Illuminations/Interactive/Cube-Nets/

◦ Stanley, R. P. Enumerative Combinatorics, Volume II, Number 62 in Cambridge Studies in Advanced Mathematics., Cambridge University Press, 1999.

◦ Turney, Peter. (1984). Unfolding the Tesseract. Journal of Recreational Mathematics, Vol. 17(1), 1984-85.

◦ Tuffley, Christopher. (2012). Counting the spanning trees of the 3-cube using edge slides. Australasian Journal of Combinatorics, Vol. 54, 187-206.