In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. Unlike the Roman surface and the cross-cap, it has no singularities (pinch points), but it does self-intersect. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. ... Werner Boy was a mathematician. ... 1901 was a common year starting on Tuesday (see link for calendar). ... The Roman surface (so called because Jakob Steiner was in Rome when he thought of it) is a self-intersecting immersion of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. ... In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. ...

To make a Boy's surface:

1. Start with a sphere. Remove a cap.
2. Attach one end of each of three strips to alternate sixths of the edge left by removing the cap.
3. Bend each strip and attach the other end of each strip to the sixth opposite the first end, so that the inside of the sphere at one end is connected to the outside at the other. Make the strips skirt the middle rather than go through it.
4. Join the loose edges of the strips. The joins intersect the strips.

Boy's surface is discussed (and illustrated) in Jean-Pierre Petit's Le Topologicon. Jean-Pierre Petit is a French scientist, member of the CNRS (Centre National de la Recherche Scientifique, National Center for Scientific Research). ...

Boy's surface was first parametrized correctly by Bernard Morin in 1978. Bernard Morin is a French mathematician, especially a topologist, born in 1931, who is now retired. ... Events January January 1 - The Copyright Act of 1976 takes effect, making sweeping changes to United States copyright law. ...

Parametrization of Boy's surface

Boy's surface can be parametrized in several ways. One parametrization, discovered by R. Bryant, is the following: given a complex number z whose magnitude is less than or equal to one, let In science, magnitude refers to the numerical size of something: see orders of magnitude. ...

so that

where X, Y, and Z are the desired Cartesian coordinates of a point on the Boy's surface. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

Property of R. Bryant's parametrization

If z is replaced by the negative reciprocal of its complex conjugate, , then the functions g₁, g₂, and g₃ of z are left unchanged. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

Proof

Let g₁′ be obtained from g₁ by substituting z with . Then we obtain

Multiply both numerator and denominator by ,

.

Multiply both numerator and denominator by -1,

.

It is generally true for any complex number z and any integral power n that

,

therefore

therefore g₁' = g₁ since, for any complex number z, The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

Let g₂′ be obtained from g₂ by substituting z with . Then we obtain

,

therefore g₂' = g₂ since, for any complex number z,

Let g₃′ be obtained from g₃ by substituting z with . Then we obtain

therefore g₃' = g₃. Q.E.D. For other meanings of the abbreviation QED, see QED. Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, that which was to be demonstrated). This is a translation of the Greek oper edei deixai which was used by many early mathematicians including Euclid and Archimedes. ...

Relating the Boy's surface to the real projective plane

Let P(z) = (X(z),Y(z),Z(z)) denote a point on Boy's surface, where . Then A spatial point is an entity with a location in space but no extent (volume, area or length). ...

but only if . What if Then

because

whose magnitude is

,

but , so that

Since P(z) belongs to the Boy's surface only when , this means that belongs to Boy's surface only if . Thus P(z) = P( − z) if , but all other points on the Boy's surface are unique. The Boy's surface has been parametrized by a unit disk such that pairs of diametrically opposite points on the perimeter of the disk are equivalent identically. Therefore the Boy's surface is homeomorphic to the real projective plane, RP². A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ... The perimeter is the distance around a given two-dimensional object. ... This word should not be confused with homomorphism. ... In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. ...

Symmetry of the Boy's surface

Boy's surface has 3-fold symmetry. This means that it has an axis of discrete rotational symmetry: any 120° turn about this axis will leave the surface looking exactly the same. The Boy's surface can be cut into three mutually congruent pieces. Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... As an abstract term, congruence means similarity between objects. ...

Proof

Two complex-algebraic identities will be used in this proof: let U and V be complex numbers, then

Re(UV) = Re(U)Re(V) − Im(U)Im(V),

Im(UV) = Re(U)Im(V) + Im(U)Re(V).

Given a point P(z) on the Boy's surface with complex parameter z inside the unit disk in the complex plane, we will show that rotating the parameter z 120° about the origin of the complex plane is equivalent to rotating the Boy's surface 120° about the Z-axis (still using R. Bryant's parametric equations given above). The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... The origin of something (from the Latin origo, beginning) is where it came from, in the sense of a physical location or a metaphysical source. ...

Let

z' = ze^{i2π / 3}

be the rotation of parameter z. Then the "raw" (unscaled) coordinates g₁, g₂, and g₃ will be converted, respectively, to g′₁, g′₂, and g′₃. In linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant under the transformation. ...

Substitute z′ for z in g₃(z), resulting in

Since e^i4π = e^i2π = 1, it follows that

therefore g₃' = g₃. This means that the axis of rotational symmetry will be parallel to the Z-axis.

Plug in z′ for z in g₁(z), resulting in

Noticing that e^{i8π / 3} = e^{i2π / 3},

Then, letting e^{i4π / 3} = e ^{− i2π / 3} in the denominator yields

Now, applying the complex-algebraic identity, and letting

we get

Both Re and Im are distributive with respect to addition, and In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

Re(e^iθ) = cosθ,

Im(e^iθ) = sinθ,

due to Euler's formula, so that This article is about the Eulers formula in complex analysis. ...

Applying the complex-algebraic identities again, and simplifying to -1/2 and to , produces

Simplify constants,

,

therefore

.

Applying the complex-algebraic identity to the original g₁ yields

Plug in z′ for z in g₂(z), resulting in

Simplify the exponents,

Now apply the complex-algebraic identity to g′₂, obtaining

Distribute the Re with respect to addition, and simplify constants,

Apply the complex-algebraic identities again,

Simplify constants,

then distribute with respect to addition,

Applying the complex-algebraic identity to the original g₂ yields

The raw coordinates of the pre-rotated point are

and the raw coordinates of the post-rotated point are

Comparing these four coordinates we can verify that

In matrix form, this can be expressed as

.

Therefore rotating z by 120° to z′ on the complex plane is equivalent to rotating P(z) by -120° about the Z-axis to P(z′). This means that the Boy's surface has 3-fold symmetry, quod erat demonstrandum. In linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant under the transformation. ... For other meanings of the abbreviation QED, see QED. Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, that which was to be demonstrated). This is a translation of the Greek oper edei deixai which was used by many early mathematicians including Euclid and Archimedes. ...

Structure of the Boy's surface

Figure 1.

Figure 1 shows a Boy's surface seen from the bottom. The nearly triangular portion is the outside surface of its "inner chamber". A Boy's surface is roughly like a bottle with three openings. Three "passageways" lead out of the inner chamber.
Reusable glass milk bottles A bottle is a small container with a neck that is narrower than the body and a mouth. ...

Figure 2.

Figure 2 shows the Boy's surface seen from near the top. The continuations of the three passageways are visible. The passageways end up intersecting into each other, forming a closed curve of double points.

Figure 3.

Figure 3 shows the Boy's surface seen from the top. About a half of the curve of double points can be seen in this figure, with the other half hidden under the folds. At the center of Figure 3 the three passageways intersect at a single triple point, which is also the point where the curve of double points intersects itself three times.

The three points of the triple point are not identically equivalent because they have different tangent planes.
In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...

Figure 4.

Figure 4 depicts the Boy's surface seen from the side. One of the surface's three "cave entrances" is shown, with a passageway coiling on top of it. A Cave Automatic Virtual Environment (better known by the recursive acronym CAVE) is an immersive virtual reality environment where projectors are directed to four, five or six of the walls of a room-sized cube. ...

Rotate Figure 4 a slight angle counterclockwise about the Z-axis and the result is Figure 5.

Figure 5.

The passageway on the left in Figure 5 leads to the one at the center which arches over the cave entrance on the right, showing a passageway at the center growing up out of the inner chamber at the bottom and slanting towards the right to form an arch.

Figure 6.

Another slight counterclockwise rotation about the Z-axis produces Figure 6. This image shows the passageway on the left pointing towards the cave entrance to the right. Also, another cave entrance is revealed: a hole on the left, under the left passageway which arches above it -under the arch of the central passageway- which feeds into the passageway which grows on the left side and arches above another cave entrance through which one may peek directly into the interior surface of the inner chamber.

Sections of the Boy's surface

The Boy's surface can be cut into six sections. Let them be called A, B, C, D, E, and F. Then sections A, C, and E are mutually congruent, and sections B, D, and F are mutually congruent.

Section A	Section B	Section C
Section D	Section E	Section F

These six sections arrange themselves into a circle, or rather a hexagon: each section corresponding to one side of the hexagon. The sections are arranged in this order: A, B, C, D, E, F -- counterclockwise around the hexagon. Each section has three sides which have been shown as orange, green, and blue. Each section can be converted through a homotopy into a triangle. The colors of the edges show how the sections are supposed to fit together. Only sides of the same color are allowed to coincide. A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... See also Orange (disambiguation) for other meanings of the word. ... Green is a colour seen commonly in nature. ... Blue (from Old High German blao shining) is one of the three primary additive colors; blue light has the shortest wavelength (about 470 nm) of the three primary colors. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... For alternative meanings, see color (disambiguation). ...

Section A′s green edge matches section D′s, B′s green edge with E, C′s green edge with F. Thus, opposite sides of the hexagon match through the green sides.

Notice that if A is rotated counterclockwise by 120°, it looks the same as C, and if it is rotated further another 120° then it looks the same as E. A similar case holds for B, D, and F.

Figure 7.

Opposite sections B and E are shown in Figure 7 joined together along their common green edge. The outside of section B becomes the inside of section E. Section B has the cave entrance under the top portion of section B which resembles an archway. The cave entrance of B leads to section E′s inner "passageway" (inner side of outer "tentacle") which eventually makes a 180° turn becoming part of the "inner chamber".

Figure 8.

Figure 8 shows another combination of opposite sections, this time sections A and D, joined along their common green edge. Section D has the cave entrance and section A has the cave's passageway into the inner chamber. There are three cave entrances in total: sections D, F, and B -- three ways to move from the outside of the Boy's surface to the inside (including the inner chamber).

Figure 9.

Figure 9 shows sections A, B, and E. Sections A and E are joined along their common green edge, and sections A and B are joined along their common blue edge. B has the cave entrance and A′s tentacle frames the top of B′s cave entrance. Every cave entrance which leads into one passageway is framed on top (like an arch) by another passageway belonging to another cave entrance. Section A intersects section E's continuation of section B′s cave entrance, such that section A becomes a "passage barrier" to section B′s cave entrance. This passage barrier can be considered to separate the cave entrance from its passageway.

Figure 10.

Figure 10 shows another view of sections A, B, and E, rotated 30° in the +z direction according to the right hand rule. The green edge of section B coincides with the green border of section E, but E seems invisible at the bottom portion of the green edge of B. This is because E curls up tightly behind B in this view.
This article or section should include material from Left handedness Handedness is an attribute of human beings defined by their unequal distribution of fine motor skill between the left and right hands. ...

Figure 11.

In Figure 11, sections A, B and E of the previous figure have been complemented with sections F, C and D, completing the Boy's surface. The cave entrance of section B remains in place, showing up as a hole pointed out by a pale blue circle surrounding it, through which one may see the interior surface of section A. This shows that the cave entrance opens directly (visually) into the "inner chamber" without being obstructed (visually) by the surface of another section. However, only a small portion of the inner chamber can be seen through this hole: this small portion is the passage barrier, which is circumscribed (and defined by) a loop of double points.

Pathways on a Boy's surface

Let a "topological ant" start out walking from the bottom of the Boy's surface (shown in Figure 12), on the outside. Let this ant walk along the green path into a cave entrance. This cave entrance is located under an archway which is like one of the tentacles of an octopus. Families 14 in two suborders, see text The octopus is a cephalopod of the order Octopoda that inhabits many diverse regions of the ocean, especially coral reefs. ...

Figure 12.

The ant goes under this archway along the dotted green path. Then the ant passes through a surface belonging to the same archway (a "passage barrier"), like a ghost passing through a wall, then walks along the inside surface of another tentacle -- a "passage" -- which feeds into the cave entrance which the ant previously walked through. Now the ant walks along the yellow path (inner surface) towards the root of the "tentacle" which leads directly into the inner chamber. The ant walks in the inner chamber towards the bottom and lands in the same point where it started, but oriented inwards. Therefore the Boy's surface, as a whole (globally), is non-orientable.

Figure 13.

Figure 13 shows the reverse case of the path shown in Figure 12. An ant starts out from the blue X on the outside of a "tentacle" then walks lengthwise along the tentacle -- along the green path -- towards another tentacle. The ant passes the other tentacle like a ghost through a wall, then finds itself in an interior surface: the inner side of a cave entrance. So it walks along the dotted yellow line until it reaches the surface of the inner chamber and the yellow path turns solid. Then the ant walks towards the orange O near the center of the inner chamber. (The ant moves from section C through its green edge into section F′s inner surface, then it walks up through section F′s blue edge into section E, and walks down section E′s inner side towards the bottom of the inner chamber.)

What if the ant were to walk along the width of a tentacle in an outward direction (away from the triple point)? See Figure 14.

Figure 14.

Then the ant, starting from section C, will cross the blue edge of section C into section D′s outer side. Then it can keep moving into section D′s cave entrance, cross section D′s green edge, move on to the top, inner side of section A (yellow path). Keep walking along the passageway (inside of tentacle) towards inner chamber at the bottom of section A.

What if the ant, starting at the blue X on the outside of the tentacle on the right (see Figure 15), were to move widthwise inbound towards the triple point?

Figure 15.

The ant starts on the outer surface of section E and walks towards its orange edge. The ant knows nothing about double or triple points; it goes right through them, like a ghost through a wall. As it crosses the orange edge, the ant walks into the outside surface of the cave entrance of section D. It can choose to walk into the cave, but let us suppose it walks the other way. It is on the higher side of D so it walks out of the archway on top of D′s cave entrance (green), crosses D′s blue edge and walks on the outer surface of section C. The ant ends on the blue O on top of another tentacle.

External links

• An interactive 3-D applet of a Boy's surface, and also some history of the surface. (http://www.uta.edu/optics/sudduth/4d/nonorientable/boys_surface/boys_surface.htm)
• Mathworld's page about Boy's surface. (http://mathworld.wolfram.com/BoySurface.html)