Inverted Boy Surface

inverted boy polar cap 001
“inverted boy polar cap” propeller-shaped polar cap of the inverted Boy surface flower-shaped polar cap of its conjugate surface. The polar disk in the domain approaches three punctures on its hemisphere, the parts nearest to the punctures get enlarged very much by the surface map.
inverted boy

This minimal surface is the image of the 6-punctured sphere. Since antipodal points of the sphere have the same image in R^3, the surface is also the image of the 3-punctured projective plane. The punctures are so called “planar ends”, which means, the surface looks outside a large ball like three pairwise orthogonal planes. Inversion of this surface in a sphere gives the famous Boy's Surface. We show the surface in three parts. The conjugate surface does not have antipodal symmetry.

inverted boy polar conj 001
inverted boy polar conj 001

Polar Cap

Inverted Boy Equator Band

inverted boy equator ba 001
inverted boy equator ba 001
inverted boy equ moeb 001
inverted boy equ moeb 001

inverted boy equator band
Inverted Boy Equator Band

The equator band in the domain starts from the equator and extends in to one hemisphere until it approaches the three punctures. As for the polar cap, the parts nearest to the three punctures get strongly expanded and are almost flat. - For the inverted Boy surface the image of the equator doubles back onto itsself so that this part of the surface is a Moebius band. _ One can follow the bands more easily in the anaglyph views.

inverted boy equator conj
Inverted Boy conjugate surface (anaglyph)
inverted boy eq moebius ana
inverted boy eq moebius ana

Inverted Boy Equator Conjugate

inverted boy equ conj 001
inverted boy equ conj 001
inverted boy equ3rd conj
inverted boy equ3rd conj
inverted boy eq conj ana
inverted boy eq conj ana
inverted boy eqconj ana2
inverted boy eqconj ana2

Inverted Boy Meridians

inverted boy 3meridians
inverted boy 3meridians
inverted boy 3meridians ana
inverted boy 3meridians ana

In the domain we can find 3 meridian bands from pole to pole which do not run through the punctures. We made them wide enough so that each comes near a puncture on both hemisspheres. For the inverted Boy surface the neigborhoods of the two domain poles have the same image: we see only polar one center. The meridian bands have therefore Moebius strips as images under the surface map. -- On the conjugate surface one can see how the (images of the) meridian bands go from polar center to polar center, each coming near a puncture on both hemisspheres.

inverted boy 3meridian conj
Inverted Boy 3 Meridian Bands
inverted boy 3merid conj ana
Inverted Boy conjugate surface (anaglyph)

Inverted Boy Meridians

inverted boy 3meridians c
inverted boy 3meridians c
inverted boy 3meridian conj c
inverted boy 3meridian conj c

inverted boy st
inverted boy st
inverted boy sw
inverted boy sw

Inverted Boys Surface

P(z) = Re ( a(z) V(z) )

where:

z = exp(u + i v)
a(z) = 1/(z^3 - z^{-3} + sqrt(5) )
V(z) = ( i ( z^2 + z^(-2) ) ,  z^2 + z^(-2) , (2 i/3) ( z^3 + z^(-3) ) )

Projective Plane Surfaces

  1. Cross-Cap Surface
  2. Boy's Surface
  3. Boy's Surface (Bryant-Kusner)
  4. Steiner Surface
  5. Inverted Boy Surface

Inverted_Boy_s_Surface.pdf