# How to Wrap the Rectangular Plane into an Infinite Alveolate Tetrahedrons

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.

I would like to share an exciting Paper Tetrahedron Folding I found with you: An A4(A or B are OK too) metric paper can be fold into multiple (2^N) interesting same shape and size tetrahedrons and the max 32 of them can combine to a large shape with the same proportion. This kind of tetrahedron named 2:root3 has 4 same isosceles triangle face, with 2 same long edges and 4 same short edges. The most changlenging one is to fold 16 same size tetrahedrons with one A4 metric paper. I named 1/2£¬ 1/4£¬ 1/8 , 1/16 and 1/32 folding. Than I have some questions and appreciate if you can give some advice: 1. How many kinds of model for the 1/2 size tetrahedrons existing ? I found 3 kinds. 2. How many kinds of model for the 1/4 size tetrahedrons existing ? I found 6 kinds. 3. How many kinds of model for the 1/8 size tetrahedrons existing ? I founds 43 kinds. 4. How many kinds of model for the 1/16 size tetrahedrons existing ? I founds 10 more kinds.

1. How many kinds of modle for the 1/32 size tetrahedrons existing ? I found 3 kinds now.

6 . I found a special method which can fold one modle of 2^N (more > than 64 )tetrahedrons. Is it other modle can be fold for the 2^N numbers?

I thought it should need computer to solve problem number 6. If it is possible,I would like to show the photoes of my multiple tetrahedrons modle family.

Sorry for my poor English.