Hans Walser, [20140111]

Regular polygon in the square

We inscribe a regular polygon in the square such that one side of the polygon is parallel to a diagonal of the square. Figure 1 depicts the situation for n = 11.

Fig. 1: Eleven-gon in the square

The angle  is close to 150ˇ/11 = 13.6364ˇ.

One may think that for  we get the limit:

ThatŐs not right. To see this, we consider the special case of regular n-gons where n is a multiple of 8 (Fig. 2 for n = 24).

Fig. 2: 24-gon in the square

In this special case itŐs easy to inscribe the polygon in the square. For the angle  we have in this special case:

Table 1 gives some examples.

 n 8 19.47122064ˇ 155.7697651ˇ 16 9.420172938ˇ 150.7227670ˇ 24 6.242752283ˇ 149.8260548ˇ 32 4.672340688ˇ 149.5149020ˇ 40 3.734283802ˇ 149.3713521ˇ 48 3.110281217ˇ 149.2934984ˇ 56 2.665117811ˇ 149.2465974ˇ 64 2.331502714ˇ 149.2161737ˇ 72 2.072157264ˇ 149.1953230ˇ 80 1.864755156ˇ 149.1804125ˇ 88 1.695106618ˇ 149.1693824ˇ 96 1.553760358ˇ 149.1609944ˇ 104 1.434177567ˇ 149.1544670ˇ 112 1.331690076ˇ 149.1492885ˇ 120 1.242875923ˇ 149.1451108ˇ 128 1.165169469ˇ 149.1416920ˇ 136 1.096609254ˇ 149.1388586ˇ 144 1.035670029ˇ 149.1364842ˇ 152 0.9811478612ˇ 149.1344749ˇ 160 0.9320797475ˇ 149.1327596ˇ 1000 0.1491172888ˇ 149.1172888ˇ 1000000 0.0001491168825ˇ 149.1168825ˇ

Tab. 1: Examples

We see, that the limit of  seems not to be 150ˇ.

In fact, the limit is in our case:

We can prove this using the rule of Bernoulli – de lŐH™pital:

In our case we have .