Golden square paper-cutouts

Let us begin with a large golden rectangle of construction paper with long side A-B of 16.2cm and short side A-D of 10cm, as shown in figure 3(I). Here, the side ratio is roughly the golden ratio. We first cut off square A-E-F-D and place it in figure 4(I). The remaining E-B-C-F of figure 3(II) is supposed to be a golden rectangle, although the ratio is only 1.61 in the present project. To be exact, B-E is 10(- 1)cm, so that we should have had

in view of relation (5). This shows that after cutting off the largest square from a golden rectangle, the leftover is again a golden rectangle.

Figure 3. Golden rectangles

Figure 4. Self-similar golden squares

Next we cut off square E-B-G-H and move it to figure 4(II). The leftover H-G-C-F of figure 3(III) is again a golden rectangle, although in actuality we only have 1.63. In this fashion, we cut off the largest squares from figure 3(III) - 3(V), and place them in figure 4(III) - 4(V). Listed in the second column of table 1 are the side lengths of five squares (I-V) of figure 4, and the scaling factors of two neighboring squares are shown in the third column. Note that the ratios are considerably different from . You may record the actual side lengths of your paper cutout squares in the last column of table 1, and compute the scaling factors to compare with the golden ratio.

  Table 1. Golden squares from rectangle of 10cm x 16.2cm

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