Odd Measures #1: Shelf encumbrance index

You can never be too thin or too rich, they say. But perhaps you can have too many books. How many is too many?

Several factors are relevant. First of all, of course, how many books you have. But whether you have too many depends also on how much space you have. Some people have enough space for all their books; each one sits neatly on its shelf with selected others, and nothing hinders the eager reader from contemplating the titles on an orderly row of spines, and plucking from amongst them the desired tome.

Low SEI.

Unfortunately, unless you are fabulously wealthy or else a “man of one book”, your books are probably not in this edenic condition. Not only may some books be shelfless, or placed ad-hockily on surfaces other than shelves, but quite possibly those that are shelved may not be entirely accessible. Some may be encumbered by others.

It is not my purpose to prescribe limits on the possession of books. Instead I aim only to provide a handy measure by which readers may decide for themselves whether their acquisitive instincts have exceeded the bounds of prudence.

To this end I offer the Shelf Encumbrance Index. In theory it is easily calculated. You have merely to determine, for each book in your collection, how many other books must be moved so as to permit that book to be available for reading. Call that its “encumbrance”. Divide the total encumbrance of all your books taken together by the number of books, and that will be the SEI. In the ideal library it is zero: no book is encumbered. Likewise the SEI of an e-library is always, sadly, zero. In Borges’ Library of Babel, on the other hand, the SEI may have been greater than zero (it can never be less), because according to Borges, a few frustrated librarians turned into vandals.*

High SEI.

In real life there will be complications. For example, in the library on the right books are not only shelved; some are stacked. Clearly the encumbrance of a book beneath the stack, if it cannot be extracted without disturbing the stack, will be at least as great as the number of books in the stack. On the other hand, the encumbrance of a book within the stack will vary. A bit of thought shows that the SEI for a stack of height n will be (n–1)/2. In yet more complicated situations empirical tests may be required.

As I said, I do not presume to determine how many books are too many. But an SEI greater than ten is clearly cause for concern. After all, you probably do want to spend more time reading your books than retrieving them (inveterate procrastinators may think otherwise).

Perhaps the easiest remedy which does not involve actually relinquishing possession of any books is to lend them. Your friends, and their friends, will soon sniff out your generosity and before long that hefty SEI will shrink fabulously.

*The alert reader will have noticed that in an infinite library, such as the Library of Babel is said to be, the SEI is not well-defined. I suggest that for infinite libraries the SEI be defined as the limit, provided it exists, of the SEIs for finite sublibraries as the diameter of the sublibraries increases without bound. This may take some time to calculate.