Topology & Mirages

Mark Evans saw this very unusual mirage over the dry estuary sands of Swansea Bay, Wales. There are multiple images of the 7km distant gas storage tanks.

Look at the sloping steps around the left-hand tank. They angle downwards from right to left. The mirage image immediately beneath it has the steps sloping in the opposite direction. The first mirage image is inverted.

We expect that.  In a usual ‘inferior mirage’, a single upside-down image is below (inferior to) the miraged object. Light waves refracted at the boundary between warm air overlain by cooler air create it.

But Mark Evans' mirage is far more interesting!  Beneath the first inverted image we see the ladder again. This time it slopes downwards in the same direction as on the real tank - this is an erect image!  Below it are four or more images alternating inverted – erect – inverted – erect …

What is happening?  We are actually seeing an unusually complete inferior mirage and the extensive, slightly undulating, warm sands have helped create it.  The undulations have tilted the boundaries between the different temperature layers.

That does not entirely explain the multiple images nor why are they so neatly alternate inverted, erect, inverted?  Answers come from topology and give unusual insight into mirage structure.

At left
is a topological representation of a 3-image inferior mirage. What is seen is printed onto a nearby transparent sphere surrounding the eye. Further out is another sphere where the actual undistorted arrow is printed life size.  The refraction and resulting curved rays are between the two spheres but we need not be concerned about temperature gradients or how or why the paths get curved. We simply look at the relations between the images on the two spheres.

Now for the magic.   The mirage, no matter how complicated, pictured on the inner
image sphere can nearly always be transformed into to the real object on the outer object sphere. To do so the inner sphere - or a copy of it - is smoothly and continuously expanded, where necessary distorting, folding or pleating (but never cutting or pasting) its rubbery surface until it meets the outer surface and the two sets of images exactly coincide. This is called transfer mapping.

The point of all this is that by mentally playing with these rubbery spheres we can derive rigorous rules about mirage structure, how their multiple images are oriented and how many images are possible.

At lower left, two folds transform the mirage into the original object. The limitations on how a surface might be folded hints at why mirage images alternate between inverted and erect but there is a more rigorous proof governing orientation.

The
degree of the transformation relates the orientation of points on the two spheres. A point on the object sphere might have corresponding points on the image sphere that are inverted or erect. The degree is the number of erect points minus the number of inverted ones. For example, at left there are two erect points and one inverted for each object sphere point. The degree is 2-1=1

It is easy (at least after being shown how*!) to prove that the degree is has to be the same all over the sphere. Once that hurdle is out of the way it follows directly that
mirages must always have a degree of one because there is nearly always part of the object sphere that is not miraged - that part has a degree of unity so the mirage itself must also have degree=1.  These abstract arguments get us to a deep practical result. Mirages like erect, inverted, erect are allowed.  Erect, erect, erect (for example) is not possible because its degree is not unity. Topology constrains the forms of mirages.

Another deep result, again easily
** derived, is the odd number theorem. Complete mirages always have an odd number of images if we include the ‘real’ erect image of the object. But what of an ‘ordinary’ inferior mirage with only one inverted image below the real one? Hence the qualification ‘complete’, in most inverted mirages the nearby ground interrupts rays that would otherwise form more images. The transfer mapping between the image and object spheres ceases to be smooth and continuous with some rays becoming tangent to the object sphere, a condition infringing the conditions for the theorems. More prosaically, the mirage ends before more images can become visible.

Mirage components can also be inconspicuous and unnoticed when they are
demagnified. The angular area of the image sphere is finite yet it must hold the many mirage images. Is there always room? Some images become compressed and less visible. Notice how in Mark's mirage the tank images are vertically compressed compared to the original.

An opposite effect occurs at
folds on the image sphere. There the image is locally magnified. Part of the mirage is vertically stretched. Fold stretching makes the green flash visible and may play a role in the vertical fantasies of the Fata Morgana.

Playful musing
*** about transparent rubbery balloons give deep insights into mirages. They tell us that upright and inverted images alternate, that complete mirages uninterrupted by ground or sea have an odd number of images, that parts can be magnified to blaze the beauty of a green flash, that parts compress into fine lines. Topology tells what mirages to search for and what we shall never see.

 * Think small rubber balloon pressed against the inside of the object sphere where there is an image and then where it is blank. ** The degree for a smooth mapping must be one. I. e there is one more erect image than there are inverted ones. Therefore the total number of images must be 2*inverted +1, an odd number. *** Some of theory was developed to explain gravitational lensing and miraging of galaxies. Thanks to Andrew Young for discussion and I am very much indebted to Walter Tape's article "The Topology of Mirages", Scientific American, 252, 120-129 (June 1985). Images ©Mark Evans, shown with permission.

 Topologically mirages can be represented by what we see pasted on a transparent inner 'image sphere' with the object pasted on a larger 'object sphere'. The mirage from the inner sphere can be transformed into the object and vica-versa. To do so the inner sphere, or a rubbery copy of it which we could call the transfer sphere, is smoothly expanded and, where needed, distorted, stretched and folded until all the mirage images overlay one another and match the object. The operation must be smooth and continuous, cuts or discontinuities are not allowed. Below: The three mirage images are made to overlay each other and be congruent to the object by folding the transfer sphere twice.
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 The sloping steps reveal the orientation of each mirage component. They alternate between inverted and erect.