$$8 \times 8 - \Delta$$

 By Watheq Al-Bnosha

Figure 1 describes $$8 \times 8 - \Delta$$
where the red line represents the diagonal and the blue lines are the tori  that make the shape of
$$8 \times 8$$.

Fig. 1

We are going to cut the diagonal and glue it, to find the new shape of $$8 \times 8 - \Delta$$ as we see in Fig. 3. But before that lets identify the tori that are going to be half before cutting and give  them  the letters $A$, $B$, $C$, and $D$. As we see in Fig. 2. 

Fig. 2

Fig. 3

Now, after we cut the diagonal. We won't have tori anymore because the diagonal is touching the ends of each torus. But we will have circles instead.  The circles are $1$, $2$, $3$, and $4$. If we connect the half circle $A$ with the half $C$, and $B$ with $D$, we will end up with $7$ circles as we see in Fig. 4.  

Fig. 4

 Now, we will prove that the two shapes in Figure 5 are homotopic to each other. We will give number $5$ to the middle circle.

                                                                             Fig. 5

The first step we will do is to give different colors to the circles from inside the shape, so we can recognize better when contracting the parts that we want as we see in Figure 6.

                                                                              Fig. 6

We are going to start contracting the purple color in circle $1$, so we get Figure 7. 

Fig.7 

By contracting the half circle $B$, we get Fig. 8

Fig. 8

Contracting the blue part in circle $4$ would give us Fig. 9

Fig. 9

Using the same method with the black part of circle $3$, and the orange part of the circle $A,C$, we get Fig. 10 & 11

Fig. 10

Fig. 11

By contracting the green part of circle $2$, circle $5$ would go inside the left over from circle $2$, and we would almost get the final shape as shown in Figure 12.

Fig. 12

Finally, we can move circle $5$ out of circle $2$ and shift it to the right and that would give us the shape we wanted to prove as we see in Fig. 13

Fig. 13