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 symmetrySymmetry 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. The three main symmetrical operations are reflection, rotation and translation. A reflection "flips" an object over a line, inverting it to its mirror image, as if in a mirror....
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Roman Surface Information

The simplest construction is as the image of a sphereA sphere is, roughly speaking, a ball-shaped object. In non-mathematical usage, the term sphere is often used for something "solid" (which mathematicians call ball). But in mathematics, sphere refers to the boundary of a ball, which is "hollow". This article deals with the mathematical concept of sphere....
[click for more] centered at the origin under the map f(x,y,z) = (yz,xz,xy). This gives us an implicit formulaA formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula) or a general relationship between quantities. A famous one is Albert Einstein's E = m × c 2 (see Special relativity)....
[click for more] of

x²y² + y²z² + z²x² − r²xyz = 0.

Also, taking a parametrization of the sphere in terms of longitudeLongitude, sometimes denoted λ, describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. Longitude is given as an angular measurement ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. Unlike latitude, which has the equator as a natural starting position, there is no natural starting position for longitude. Therefore, a reference meridian had to be chosen. While British cartographers had long used the Greenwich meridian in L...
[click for more] (θ) and latitude (φ), we get parametric equations for the roman surface as follows:

x = r² cos θ cos φ sin φ

y = r² sin θ cos φ sin φ

z = r² cos θ sin θ cos² φ

The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each axis which terminate in pinch points. The entire surface has tetrahedralA tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral," and is one of the Platonic solids. ...
[click for more] symmetryThe symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. The article on group theory also contains an explanation of the concept. In Euclidean geometry, discrete symmetry groups come in two types: finite point groups, which include only rotations and reflections, and infinite lattice groups, which also include translations and glide reflections. There are also continuous symmetry groups, which are Lie groups....
[click for more]. It is a particular type (called type 1) of Steiner surfaceIn geometry, the Steiner surfaces are certain self-intersecting embeddings (that is to say, immersions) of the real projective plane into three-dimensional space. More particularly, they are linear projections of a six-dimensional embedding called the Veronese surface, which is the image of an ordinary 2-sphere centered at the origin under the map...
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Derivation of implicit formula

The points on a sphere have coordinates

.

To these points, apply a transformation T defined as

so that the transform of the sphere has points whose coordinates are

From equation (1) it follows that

Equations (2), (3), and (4) combine through multiplication, producing these next three equations:

Equations (5), (6), and (7) combine through addition, producing

Factor out the x² y² on the right side,

Now factor out (r² − x² − y²) on the right side,

Multiplying the three components of equation (1) yields

therefore equation (8) can be restated as

where r is the radius of the original sphere; the Roman surface does not have a radius.

Derivation of parametric equations

Let a sphere have radius r, longitude φ, and latitude θ. Then its parametric equations are

Then, applying transformation T to all the points on this sphere yields

which are the points on the Roman surface. Let φ range from 0 to 2π, and let θ range from 0 to π/2.

Relation to the real projective plane

The sphere, before being transformed, is not homeomorphic to the real projective plane, RP². But the sphere centered at the origin has this property, that if point (x,y,z) belongs to the sphere, then so does the antipodal point (-x,-y,-z) and these two points are different: they lie on opposite sides of the center of the sphere.

The transformation T converts both of these antipodal points into the same point,

If this were true for only one or small subset of points of the sphere, then these points would just be double points. But since it is true of all points, then it is possible to consider the Roman surface to be homeomorphic to a "sphere modulo antipodes", S² / (x~-x), i.e. a sphere whose antipodal points are equivalent. The real projective plane is known to be homeomorphic to a sphere modulo antipodes, therefore the Roman surface is homeomorphic to RP².

Structure of the Roman surface

The Roman surface has four bulbous "lobes", each one on a different corner of a tetrahedron.

A Roman surface can be constructed by splicing together three hyperbolic paraboloidsIn mathematics, a paraboloid is a quadric, a type of surface in three dimensions, described by the equation: (elliptic paraboloid), or...
[click for more] and then smoothing out the edges as necessary so that it will fit a desired shape (e.g. parametrization).

Let there be these three hyperbolic paraboloids:

• x = y z,
• y = z x,
• z = x y.

These three hyperbolic paraboloids intersect externally along the six edges of a tetrahedron and internally along the three axes. The internal intersections are loci of double points. The three loci of double points: x = 0, y = 0, and z = 0, intersect at a triple point at the originT...
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For example, given x = y z and y = z x, the second paraboloid is equivalent to x = y / z. Then

and either y = 0 or z² = 1 so that . Their two external intersections are

• x = y, z = 1;
• x = -y, z = -1.

Likewise, the other external intersections are

• x = z, y = 1;
• x = -z, y = -1;
• y = z, x = 1;
• y = -z, x = -1.

Let us see the pieces being put together. Join the paraboloids y = x z and x = y z. The result is shown in Figure 1.

Figure 1.

The paraboloid y = x z is shown in blue and orange. The paraboloid x = y z is shown in cyan and purple. In the image the paraboloids are seen to intersect along the z = 0 axis. If the paraboloids are extended, they should also be seen to intersect along the lines

• z = 1, y = x;
• z = -1, y = -x.

The two paraboloids together look like a pair of orchids Orchids (Orchidaceae family) are among the largest and most diverse of the flowering plant (angiosperm) families, with over 800 described genera and 25,000 (some sources give 30,000) species, and perhaps another 60,000 hybrids and varieties produced by horticulturalists. The Kew checklist "World Checklist of Orchids" gives about 24,000 accepted names. About 800 new species are added each year. There are more orchid species than all vertebrates combined, excluding bony fishes. Orchids, through their floral ...
[click for more] joined back-to-back.

Now run the third hyperbolic paraboloid, z = x y, through them. The result is shown in Figure 2.

Figure 2.

On the West-Southwest and East-Northeast directions in Figure 2 there are a pair of openings. These openings are lobes and need to be closed up. When the openings are closed up, the result is the Roman surface shown in Figure 3.

Figure 3. Roman surface.

A pair of lobes can be seen in the West and East directions of Figure 3. Another pair of lobes are hidden underneath the third (z = x y) paraboloid and lie in the North and South directions.

If the three intersecting hyperbolic paraboloids are drawn far enough that they intersect along the edges of a tetrahedron, then the result is as shown in Figure 4.

Figure 4.

One of the lobes is seen frontally -- head on -- in Figure 4. The lobe can be seen to be one of the four corners of the tetrahedron.

If the continuous surface in Figure 4 has its sharp edges rounded out -- smoothed out -- then the result is the Roman surface in Figure 5.

Figure 5. Roman surface.

One of the lobes of the Roman surface is seen frontally in Figure 5, and its bulbous -- balloon-like -- shape is evident.

If the surface in Figure 5 is turned around 180 degrees and then turned upside down, the result is as shown in Figure 6.

Figure 6. Roman surface.

Figure 6 shows three lobes seen sideways. Between each pair of lobes there is a locus of double points corresponding to a coordinate axis. The three loci intersect at a triple point at the origin. The fourth lobe is hidden and points in the direction directly opposite from the viewer. The Roman surface shown at the top of this article also has three lobes in sideways view.

One-sidedness

The Roman surface is non-orientable, i.e. one-sided. This is not quite obvious. To see this, look again at Figure 3.

Imagine an ant on top of the "third" hyperbolic paraboloid, z = x y. Let this ant move North. As it moves, it will pass through the other two paraboloids, like a ghost passing through a wall. These other paraboloids only seem like obstacles due to the self-intersecting nature of the immersion. Let the ant ignore all double and triple points and pass right through them. So the ant moves to the North and falls of the edge of the world, so to speak. It now finds itself on the northern lobe, hidden underneath the third paraboloid of Figure 3. The ant is standing upside-down, on the "outside" of the Roman surface.

Let the ant move towards the Southwest. It will climb a slope (upside-down) until it finds itself "inside" the Western lobe. Now let the ant move in a Southeastern direction along the inside of the Western lobe towards the z = 0 axis, always above the x-y plane. As soon as it passes through the z = 0 axis the ant will be on the "outside" of the Eastern lobe, standing rightside-up.

Then let it move Northwards, over "the hill", then towards the Northwest so that it starts sliding down towards the x = 0 axis. As soon as the ant crosses this axis it will find itself "inside" the Northern lobe, standing right side up. Now let the ant walk towards the North. It will climb up the wall, then along the "roof" of the Northern lobe. The ant is back on the third hyperbolic paraboloid, but this time under it and standing upside-down. (Compare with Klein bottleIn mathematics, the Klein bottle is a certain genus-1 non-orientable surface, i.e. a surface (a two-dimensional topological space), for which there is no distinction between the "inside" and the "outside" of the surface. The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It is closely related to the Möbius strip and embeddings of the real projective plane such as Boy's surface....
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Double, triple, and pinching points

The Roman surface has four "lobes". The boundaries of each lobe are a set of three lines of double points. Between each pair of lobes there is a line of double points. The surface has a total of three lines of double points, which lie (in the parametrization given earlier) on the coordinate axes. The three lines of double points intersect at a triple point while lies on the origin. The triple point cuts the lines of double points into a pair of half-lines, and each half-line lies between a pair of lobes. One might expect from the preceding statements that there could be up to eight lobes, one in each octant of space which has been divided by the coordinate planes. But the lobes occupy alternating octants: four octants are empty and four are occupied by lobes.

If the Roman surface were to be inscribed inside the tetrahedron with least possible volume, one would find that each edge of the tetrahedron is tangent to the Roman surface at a point, and that each of these six points happens to be a Whitney singularityIn mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. See singularity theory for general discussion of the geometric theory, which only covers some aspects....
[click for more]. These singularities, or pinching points, all lie at the edges of the three lines of double points, and they are defined by this property: that there is no plane tangent to surface at the singularity.

See also: Boy's surface.

External links

• An interactive 3D applet of a Roman surface, and also a discussion of the surface. (http://www.uta.edu/optics/sudduth/4d/nonorientable/steiners_roman_surface/steiners_roman_surface.htm)
• Ashay Dharwadker, Heptahedron and Roman Surface, Electronic Geometry Models, 2004. (http://www.eg-models.de/models/Surfaces/Algebraic_Surfaces/2003.05.001/)

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