f5102 | JP-Petit.info

ANALYSIS OF RESULTS.

Figure 3 shows the periodic functions A(m) and B(m). B is simply phase-shifted relative to A.

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Figure 3.

Using an "Apple-II" microcomputer, we plotted a view of Boy's surface showing the elliptic meridians intersecting the single pole.

Now consider the cross-sections Z = constant. Their equations follow directly from that of the surface. These cross-sections are drawn in Figures (5a) to (c). All figures exhibit ternary symmetry, as can be seen. The first three cross-sections display inflection points. These slight irregularities are the traces of cusp singularities appearing in this region before coefficient adjustment. In Figure (5j), three sealed points are found. The two circles embedded in this figure (5j) have neighborhoods that are Möbius bands, twisted three times by half-turns relative to the horizontal plane z = constant.

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Figure 4. Meridional lines (E m) of Boy's surface plotted using an "Apple II".

Below are improved illustrations compared to those accompanying the original note in the Comptes Rendus of the Académie des Sciences de Paris:

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Fig. 5a

Fig. 5b

Fig. 5c

Fig. 5d

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Fig. 5e

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Fig. 5f

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Fig. 5g

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Fig. 5h

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Fig. 5i

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Fig. 5j

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Fig. 5k

Fig. 5l

Figures 5a to 5l

The cross-section (5g) passes through the triple point of the surface. Cross-sections (5f), (5j), and (5m) correspond to limiting cases where changes occur in the way curve arcs are joined.

In Figure (5i), we have indicated the sealed points by:

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