A weekend with Wolfram Alpha

I’ve spent a little time over the past weekend throwing various expressions at Wolfram Alpha, seeing what I can do with it, and what it can do for me. It certainly has some very fine aspects and can do some things extremely well. But I kept coming up against a wall – finding things I assumed it should be able to do, but can’t.

Mathematics

Given that Mathematica provides the computational engine to Wolfram Alpha, we would expect that its mathematical ability to be second to none. So I threw a few expressions at it: first an integral

which I entered as

integrate 1/sqrt(1-x^3)


and which was answered immediately, with a handsome result involving elliptic functions. For free, I got a few plots as well.

Then a differential equation

as

y'=x^2+y^2,y(0)=1


and which was identified as a Riccati equation, and solved precisely. I would have liked to find the series solution for this last equation, but I don’t know how to do this using W|Alpha.

For fun, I tried a large matrix expression, in particular to evaluate the weights of the Newton-Cotes fourth-order rule, by means of

as

(inv {{1,1,1,1,1},{1,2,3,4,5},{1,4,9,16,25},{1,8,27,64,125},{1,16,81,256,625}}).{{4},{(5^2-1)/2},{(5^3-1)/3},{(5^4-1)/4},{(5^5-1)/5}}


As a super-calculator, then, W|Alpha is completely and utterly wonderful, and in giving us lowly mortals a window into the computational and symbolic power of Mathematica, W|Alpha is a great service indeed.

I started getting some tetchiness when I attempted to experiment with some slightly more abstruse topics. I decided to try to find the primitive root of the next prime after one trillion:

primitive root next prime one trillion


The system choked on this. I found a result first by finding the prime

next prime one trillion


(which turned out to be 1000000000039), and then copying this into

primitive root 1000000000039


I then thought of calculating a few knot groups, starting with the knot group of the (5,2) torus knot. There seemed to be no way of doing this; certainly entering

5 2 torus


gives a lot of information about the knot (some nice plots, various associated polynomials), but not its group. Attempts to find other topological invariants (homotopy groups, homology groups) also ended in disappointment. I don’t know whether Mathematica can do this or not, but it seems that W|Alpha can’t, as yet. When I entered

knot group


I got a sheaf of information about a New York company. This was severely disappointing. I had hoped that from an engine based on mathematical software, “group” would be interpreted in its mathematical, algebraic sense, but apparently not.

There doesn’t seem to be a way of saving a result (with a variable) for use in a subsequent calculation. But you can always cut and paste. It seems then, that the amount of Mathematica knowledge and functionality released into W|Alpha is limited. And although it may be useful for quick results, it can no way take the place of a proper computer algebra system.

Geography

In quick order I found the distances between my home town of Melbourne, Australia, and other cities; each time accompanied with a map indicating the shortest route between them. However, my attempts to find the city furthest from Melbourne couldn’t be done:

furthest city from Melbourne


returned the standard error: “Wolfram|Alpha isn’t sure what to do with your input.” I tried with different words, but to no avail.

Life Sciences

I tried a few biological queries:

smallest mammal

obligingly returned some information about the Etruscan shrew (which is the smallest by weight), but nothing about Kitti’s Hog-Nosed Bat, which is in fact the smallest mammal by length. This seemed to be a lucky dip for me; lots of other queries came to nothing:

largest land mammal
longest jellyfish
heaviest snake

Interestingly, the query

largest snake

returned the reticulated python, which is the world’s longest snake, but in fact the largest snake (by bulk) is the anaconda. When I entered, for the infamous box jellyfish,

chironex fleckeri

I received its biological breakdown (kingdon, phylum, class, order etc), but not its common name, nor any pictures, nor any information about the animal’s behaviour.

Music

According to the instructions, you can enter the name of a song, and indeed

unchained melody

returned information about the song being released in July 1965 by The Righteous Brothers. However, a newer song

how to save a life

(by The Fray; one of my daughter’s favourite songs) returned nothing. Also, by what seems a weird oversight, W|Alpha knows nothing about classical music; the searches:

haffner serenade
eroica symphony
the marriage of figaro

all returned nothing. There’s no way of searching by Koechel numbers for Mozart works, or by BWV numbers for Bach’s works. W|Alpha does know a little about tuning, scales and intervals, but nothing about temperaments:

werkmeister 3

returned nothing.

Conclusions

So how good, then, is Wolfram Alpha? It is touted as a “knowledge engine” – its aims are quite different to Google’s – and it claims to be able to give answers to queries from many disciplines. I found it vaguely annoying, with occasional flashes of amazing brilliance (especially in its mathematics). Maybe as its database grows, and more “intelligence” is added to its system, I will indeed be able to find what city is furthest from Melbourne, or when the next total solar eclipse will be visible in Edinburgh, or what the knot group is for the (5,2) torus knot.