60 degree triangle preserving matrices.

The 120 degree triangle preserving matrix

$$\mathbf{M}=\begin{pmatrix} 4 & 3 & 6\\ 3 & 4 & 6\\ 4 & 4 & 7 \end{pmatrix}$$

we found in the previous section, can be transformed into four 60 degree triangle preserving matrices. To do so, we will need the following identity,

$$x^2+y^2+xy=(x+y)^2+y^2-(x+y)y=x^2+(x+y)^2-x(x+y).$$ (40)

We have $a^2+b^2-ab=(-a)^2+b^2+(-a)b=c^2$ . Then using $\mathbf{M}$ ,

$$\begin{pmatrix} -a&b&c \end{pmatrix}\begin{pmatrix} 4 & 3 & 6\\ 3 & 4 & 6\\ 4 & 4 & 7 \end{pmatrix} = \begin{pmatrix} u &v & w \end{pmatrix}$$

where $u^2+v^2+uv=w^2$ . However, from equation (40), $(u+v)^2+v^2-(u+v)v=w^2$ . Hence

$$\begin{pmatrix} -a&b&c \end{pmatrix}\begin{pmatrix} 4+3 & 3 & 6\\... ...nd{pmatrix}\begin{pmatrix} -7 & -3 & -6\\ 7 & 4 & 6\\ 8 & 4 & 7 \end{pmatrix}$$

where $a^2+b^2-ab=c^2$ is a 60 degree triangle. Similarly, if $a_1^2+(-b_1)^2+a_1(-b_1)=c_1^2$ , then

$$\begin{pmatrix} a_1&b_1&c_1 \end{pmatrix}\begin{pmatrix} 7 & 3 &6... ...-6\\ 8 & 4 & 7 \end{pmatrix}= \begin{pmatrix} u_1+v_1 &v_1 & w_1 \end{pmatrix}$$

where $a_1^2+b_1^2-a_1 b_1=c_1^2$ and $(u_1+v_1)^2+v_1^2-(u_1+v_1)v_1=w_1^2$ .

Let

$$\boxed{\mathbf{S}_1=\begin{pmatrix} 7 & 3 &6\\ -7 & -4 & -6\\ 8... ...f{T}_1= \begin{pmatrix} -7 & -3 & -6\\ 7 & 4 & 6\\ 8 & 4 & 7 \end{pmatrix}.}$$

Then, clearly,

$$\boxed{\mathbf{S}_2=\begin{pmatrix} 4 & 7 &6\\ -3 & -7 & -6\\ 4... ...bf{T}_2= \begin{pmatrix} -4 & -7 & -6\\ 3 & 7 & 6\\ 4 & 8 & 7 \end{pmatrix}}$$

are also 60 degree triangle preserving matrices.

For example, since $8^2+3^2-(8)(3)=7^2$ ,

$$\begin{pmatrix} 8 & 3 & 7 \end{pmatrix}\mathbf{S}_1= \begin{pmatr... ...& 3 & 7 \end{pmatrix}\mathbf{S}_2= \begin{pmatrix} 51 & 91 & 79 \end{pmatrix},$$

$$\begin{pmatrix} 8 & 3 & 7 \end{pmatrix}\mathbf{T}_1= \begin{pmatr... ... 8 & 3 & 7 \end{pmatrix}\mathbf{T}_2= \begin{pmatrix} 5 & 21 & 19 \end{pmatrix}$$

where

$$91^2+40^2-(91)(40)=79^2,\qquad 51^2+91^2-(51)(91)=79^2,$$

$$21^2+16^2-(21)(16)=19^2,\qquad 5^2+21^2-(5)(21)=19^2.$$

Download these 3 sections in pdf form. triples8.pdf

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