On Wednesday night it suddenly occurred to me that a particular book in my Borges collection had gone missing. It was 2:30. I went to the den where I keep my books in Spanish and found Jorge Luís Borges  
El Hacedor (an Alianza Emecé) missfiled.  It was easy to find as the cover has an illustration that incorporates the title within a Möbius strip. Those who might not know should know that the strip is an instance of one-sided surface. And if you further look you note that a Möbius strip is of the same shape as the mathematical symbol for infinity, ∞.

When I put the book in its place I noticed a little one I had not read for some years. El Lenguaje de Buenos Aires (also published by Emecé) which contains essays about the language spoken in Buenos Aires, written by Jorge Luís Borges and José E. Clemente. In the first essay by Borges, El Idioma de los Argentinos, a printed version of a lecture he gave in 1927 in Buenos Aires I found this:

Sospecho que la palabra infinito fue alguna vez una insípida equivalencia de inacabado; ahora es una de las perfecciones de Dios en la teología y un discutidero en la metafísica y un énfasis popularizado en las letras y una finísima concepción renovada en las matemáticas –Russell explica la adición y multiplicación y potenciación de números cardinales infinitos y el porqué de sus dinastías casi terribles- y una verdadera intuición al mirar al cielo.

My translation into English is:

I suspect that the word infinity was at one time an insipid equivalent of the unfinished. It is now one of the perfect attributions of the God of theology and a centre for discussion in philosophy and popularized with emphasis in arts and letters. It is a fine concept, renewed often in mathematics – Russell [Bertrand] the addition, multiplication, and the powering of infinite cardinal numbers and whence the terrible place they came from – and of a true intuition that comes upon looking up into the sky.

I thought about that marvelous paragraph (the one in Spanish and not my poor translation) and combined that with the image of the Möbius strip and came up with this: 


Before the proliferation of GPS devices for cars and phones most of us resorted to using maps to find places and or used our memories and some logic to get to our destination. This ability to find a place on one’s own will be surely lost as we come to depend more and more on location technology.

Before cameras had auto focus lenses, photographers had to focus manually. Before the invention of single lens reflex cameras, during the era of the uncoupled rangefinder cameras, photographers like Cartier-Bresson would focus using a little dark window that matched two images (let’s say Uncle Billy’s face). When the two faces were one the camera was focused. Then Cartier-Bresson would look into another window to frame his shot.

In all those situations and the more primitive one of guessing a distance and manually setting a lens to that distance the photographer made a choice. It could be a stupid choice or an intelligent one based on experience.

All that is now almost history. The auto-focusing cameras have just about taken the decision making choice on where to focus out of a photographer’s hand/eyes.

That does not mean that the curious photographer should not attempt to find out exactly what all those focusing points in the camera’s viewing screen are doing.

Consider the problem of attempting to photograph your Uncle Billy with the Grand Canyon behind him. Where do you focus?

Optics and the laws of optics since Isaac Newton discovered them are up to this point set in stone. Whenever we look at anything with our eyes, with a camera (a still one or a movie one), with binoculars or any other device there will always be one plane of sharp focus.

This plane of focus, be it near or far will have a distance behind it and a distance in front of it that will be in acceptable focus. These two planes behind and in front are called depth of field. That plane will be approximately parsed at 1/3 in front and 2/3 in back. That depth of field zone will be narrow if you are up close and wider as you move away.

When you look at that north rim of the Grand Canyon, that 2/3 depth of field beyond is of no consequence. The Grand Canyon is at infinity (∞). That can be pretty far! But beyond infinity (∞) there is more of it.

So going back let’s say you are shooting a tight portrait of Uncle Billy. He has a big nose. Your problem is that you want the tip of his nose sharp and also his ears. So knowing about that 1/3 in front and 2/3 in back you might focus on the eyes (the eyelashes). All things considered if you are using the right f-stop in your camera the ears and the nose will be sharp. This is because you are moving that depth of field zone so that the plane of sharp focus is by the eyes.

Now Uncle Billy is at the Grand Canyon. You want him in focus and the Canyon behind in focus. Where do you focus your camera?

That place has the complicated name of hyper focal distance.

Note in the scan of my manual focus Nikon 35mm lens. On the left you will note:

1. Where the ∞ is lies, to the left of it is a small vertical red line. That vertical red line (faded because this lens is very used and very old) matches the colour of the number 22 which happens to be the smallest aperture of that lens. Just like when you squint your eyes to read the sign on the bus, when you close a lens (you squint it) you increase the zone of depth of field. On the opposite side (to the right) of that vertical red line you might note that it lies a little to the left of the number 1 representing 1 metre. That scale is telling you that if you have your lens set as is (Uncle Billy is now about 2 metres away (a big black dot on the lens), Uncle Billy will be in focus and so will the Grand Canyon. Or everything from 1 metre away (Uncle Billy) to the Grand Canyon (a photographic lens infinity away (∞) will be in focus. This is the hyper focal distance of a 35mm lens (whatever brand you want it will be the same) at f-22.

What is interesting is that your lens is focusing on a spot of no particular importance, a metre behind Uncle Billy. Putting it in another way, to focus on what you want in focus you have to focus elsewhere

How would your $6000 Canon DSL handle that problem?

Thus at 2:30 in the morning a couple of Borges books, a Nikon lens, a couple of sketches have all come together around the theme and or concept of infinity. Which brings me to the idea that by its very definition of infinity, infinity has to be eternal. Both difficult terms are linked in that Zeno paradox of Achilles and the tortoise. Achilles can never reach the destination as he has to achieve half of it, and half again.  And that can’t be and yet we know that Achilles will sprint and win the race in spite of our philosophical musings to the contrary.

It is interesting for me to know that thanks to Leibniz and Newton (who independently discovered the calculus) we know about incremental changes in slopes (in differential calculus) and how that led in the 20^th century to hyperbolic paraboloid roofs in modern buildings and that the source is a straight line from a point of swivel that has an ever changing slope so that the individual lines, when taken together form a graceful sweep of a roof. That differential calculus and Newton’s discovery of the laws of gravitation gave us accurate ballistics is a tragedy that we cannot avoid or find a remedy for.

Integral calculus with the elegant sweeping of a right angle triangle from its base (at the right angle) for 360 degrees will give us the volume of a cone of that triangle’s height is one of those moments of my life that I will never forget. As I find all these many (and there are many) citations of the infinite, of the eternal in Borges I feel so lucky to be alive.

And luckier still to be able to read the man in the language of Buenos Aires.