Tetrahedron Kites

Tetrahedron Kites And the mathematics embedded within. By Art Mabbott Seattle Schools and Park City Math Institute

Level Triangular Numbers 1st Difference 2nd Difference 0 0 0 1 1 1 1 2 3 2 1 3 6 3 1 4 10 4 1 5 15 5 1 6 21 6 1 7 28 7 1 8 36 8 1 N Must be a Quadratic

Level Triangular Numbers Factoring Tri #’s Twice Tri #’s Factoring Twice Tri #’s 0 0 0*0 0 0*1 1 1 1*1 2 1*2 2 3 1*3 6 2*3 3 6 2*3 12 3*4 4 10 2*5 20 4*5 5 15 3*5 30 5*6 6 21 3*7 42 7 28 4*7 56 8 36 4*9 72 N N*(N+1) So… N*(N+1)/2

Tetrahedron Numbers 3rd Difference 1st Difference 2nd Difference 0 0 0 0 1 1 1 1 1 2 4 1 3 2 3 10 1 6 3 4 20 1 10 4 5 35 1 15 5 6 56 1 21 6 7 84 1 28 7 8 120 1 36 8 9 165 1 9 45 10 220 1 10 55

Tetrahedron Numbers Three times Six times Factors Twice 0 0 0 0=0*1*2 0 1 1 3 6=1*2*3 2 2 4 12 24=2*3*4 8 3 10 20 30 60=3*4*5 4 20 40 60 120=4*5*6 5 35 70 105 210=5*6*7 6 56 112 168 336=6*7*8 7 84 168 252 504=7*8*9 8 120 240 360 720=8*9*10 N [N*(N+1)*(N+2)]/6 N*(N+1)*(N+2)

Let’s Make the Kites

1 Triangle n 4n n 2 2 3 Number Length Area Total of of a Perimeter of a Surface Total Volume Straws Side Face Area 1 1 3 4 1 Tetrahedron 4 2 2 6 16 8 9 3 3 9 36 27 16 4 4 12 64 64 25 5 5 15 100 125 6 6 18 36 144 216 7 7 21 49 196 343 8 8 24 64 256 512 9 9 27 81 324 729 10 10 30 100 400 1000 20 20 60 400 1600 8000 100 100 300 10000 40000 1000000 … n 3n

3 n 3 2 3 3 2 (n +3n +2n)/6 (4n -4n)/6 (n -3n +2n)/6 Down Tetrahedrons Total Volume Number of Straws Up Tetrahedrons Octahedron 1 1 - - 1 4 = 1 + 3 2 1(4) - 8 10 = 1 + 3 + 6 3 4(4) 1 27 20 = 1 + 3 + 6 + 10 4 10(4) 4 64 35 = 1 + 3 + 6 + 10 + 15 5 20(4) 10 125 56 6 35(4) 20 216 84 7 56(4) 35 343 120 8 84(4) 56 512 165 9 120(4) 84 729 10 220 165(4) 120 1000 20 1540 1330(4) 1140 8000 100 171700 166650(4) 161700 1000000 … (n)(n+1)(n+2)/6 (4)((n-1)(n)(n+1)/6) ((n-2)(n-1)(n))/6

To Contact me: Art Mabbottart@mabbott.orgTo contact me:(206) 605-7393http:mabbott.org/mathguy.htmhttp:mabbott.org/tetprep.htm

Tetrahedron Kites