Sunday, December 12, 2021

Did Lane Hemaspaandra invent the Fib numbers?

 (I abbreviate Fibonacci by Fib throughout. Lane Hemaspaandra helped me with this post.) 

We all learned that Fib invented or discovered the Fib Numbers:


f_1=1, and

for all n\ge 2, f_n = f_{n-1} + f_{n-2}.

We may have learned that they come up in nature (NOT true, see here) or that they were important in mathematics (questionable--see this blog post here which says no, but some comments give good counterarguments). You also learned that Fibonacci was the mathematician who first studied them. Also not true!  This one surprised me. 

1) I came across this blog post: here that says they were invented by Hemachandra first. Wow--I then recalled that Lane Hemaspaandra's birth surname  was Lane Hemachandra (he married Edith Spaan and they are now both Hemaspaandra). So naturally I emailed him to ask how a 20th-century person could invent something earlier than 1170. He told me a picture of him in the basement ages while he stays young. 

2) It would be nice to say OH, let's call them Hemachandra numbers (would that be  easier than convincing the world to use tau instead of pi,? See The Tau Manifesto) and let students know that there were people other than Europeans who did math back then. But even that story is not as simple as it seems. Lane emailed me this article: here that tells the whole muddled story. (In what follows I leave out the accents.)

Virahanka seems to be the formulator of the Fib recurrence, though not quite the numbers. His motivation was Sanskrit Poetry.  He did this between 600 and 800 AD.

Gopala, in work prior to 1135, was aware of Virhanka's work. In particular he know about the inductive rule. But he also set the initial values and generated numbers, so he was the first to have the sequence we now call the Fib numbers. His motivation was Sanskrit Poetry. 

Hemachandra in 1150 also formulated them, independently.  His motivation was Sanskrit poetry.

(I will learn some Sanskrit poetry the next time I teach Discrete Math so I can give the students an application of the material!)

So does Virhanka win? Hardly:

Acarya Pingala's writings from the 5th or 6th century BC (YES- BC!) indicate that he knew about the Fib numbers in the context of (you guessed it!) Sanskrit poetry. 

3) I would support changing the name of the Fib Numbers to the Pingala numbers. This is both good and bad news for Lane:

a) Bad news in that he does not get a sequence of number that shares his pre-marriage name.

b) Good news in that if I settled on Hemachandra numbers then Lane and Edith would have to decide if 0 or 1 or 2 of them want to change their name to Hemachandra. I would guess not--too complicated. Plus one name change in a life is enough. 

4) The story (especially  the articles I pointed to) shows just how complicated history can get. Even a straightforward question like:

Who first formulated the Fib Numbers?

might not have a well-defined answer. Perhaps this is  the wrong question since if people formulate the concept independent of each other, they should all get some credit.  Even if the authors are 1000 years apart. 

Side note: Independent Discovery may be harder to assert now since, with the web,  Alice could have seen Bob's paper so it may be hard to call Alice's discovery independent. As I have mentioned before on this blog, my students have a hard time with the notion of Cook and Levin coming up with NP-completeness independently  since  surely one would have posted it and the other would have seen it. An era before posting was possible! Unimaginable to them. Sometimes even to me. 


  1. Fibonacci sequences are indeed useful and interesting. What about a Fibboacci complexity class.

  2. To analyze the Euclidean GCD algorithm.