how to wrap the rectangular plane into an infinite alveolate
TRANSCRIPT
How to Wrap the Rectangular Plane into an Infinite Alveolate Tetrahedrons
Haisheng Liang Wenwu Chang
Shanghai Conference on Algebraic Combinatorics
2012. 19th Aug.
Abstract
On Euclidean space, the A4 size paper as well as other Silver Rectangle with aspect ratio √ 1: √ 2 , can be wrapped into 2^N and 2(2N+1) ^ 2 congruent tetrahedrons for each face. The rectangular area equals to its wrapped surface area of all tetrahedron. In other words, the rectangle can be bent and folded into 4N same unit of an isosceles triangle small pieces to form the N of the similar tetrahedron. This kind of tetrahedron is that surrounded by the similar one where space can be filled with this isosceles tetrahedron. There are only two kinds of edge and its ratio is √ 3 :2 . When wrapping is near to the infinite 2 ^ N and 2(2N+1) ^ 2 congruent tetrahedron, it is found that there are three methods.
The first one is to expand on the XY plane where the tetrahedron rows misplaced tiling method. The second one is to expand around the central axis on the XYZ three-dimensional method. This method forms two types of "honeycomb", tetrahedron and octahedron, where the former one is larger than the later one. The third method is to expand around the central axis on the XYZ three-dimensional method. It forms only one type of "honeycomb" which is tetrahedron without gap. When N is natural number such as 1,2,3,4… and A4 size paper wrapped into 2 ^ N and 2(2N+1) ^ 2 congruent tetrahedron, it would be different models on the shapes of plural kinds, based on polyhedral Euler's formula. How to find these different models? With the increase of N , would the number of models be the convergence or the divergence? Many unknown interesting habitude is to be explored.
Abstract
1st Method’s element by ¾ A4 Paper 1 : 3 √ 2/4
With 3x4 grid ¾ Paper fold to the basic model
Peek
Valley
Type A
Type B
¾ Paper fold to the basic model
Peek
Valley
Element Type A Type B
Type A 4 elements
Type B 4 elements
Type A Type B
Mirror
64 tetrahedrons expanding
Second Method Use A4 Paper
Peek
Valley
¼ Paper fold to the basic element
Top view
A A’ B B’
A and A’ are both on the same plane
B and B’ are both on the same plane
Two elements Connect by Edges
¼ Tetrahedra
½ Tetrahedra
Octahedron
½ Octahedron
½ Octahedron
½ Tetrahedra
Margin than Change Edge Out
+ = =>
Top view
Two Edges Connect together
+ = =>
+
<=
+ =
=>
3rd Method element by ¾ A4 Paper 1 : 3 √ 2/4
with 3x4 grid ¾ Paper fold to the basic model
View it from 4 directions 1 3 1 & 2 in a face, 3 & 4 in a
face , 1,2 _I_ 3,4 2 4
Eye near middle far
Y Z X
Expand direction Y
A B
A B
A B
A B
Expand direction Z
X-Axel touch
Expand direction X
3x4 => 6x8……..12x16….24x32…..
32 Tetrahedrons Example
