Can amateurs still make discoveries in mathematics?

In a maths forum someone posted a link to a Spiegel article (in German) about crackpots. This elicited a predictable stream of comments with one group ridiculing the crackpots and another complaining about the arrogance of the academics.

The interesting point of view was that all the low hanging fruit have been discovered. Amateurs have no chance of success because there are more professional mathematicians in the world than ever before and all the progress is at the outer fringes of knowledge which require many years of study to get to.

How does that square with my rotational numbers?

To reiterate, I don’t see them as a radical new discovery, but a reinterpretation of existing knowledge.

Rotational numbers are not an advanced concept at the cutting edge of mathematical research, but a basic number set that, in my view, was passed over in the relentless drive towards abstraction.

The rules and laws of rotational numbers have already been discovered: as cyclotomic fields. The difference is that in my system the unity roots are not complex numbers, but fundamental constants,like 1,-1 or i. The number i is the order 4 rotational constant, which makes complex numbers a subset of rotational numbers rather than the one underlying set. This is much more general and elegant, but at the price not having general multiplication between rotational numbers.

You could argue that someone would have pointed this out before, if it was really a better view. Are there precedents where mathematicians are attached to a viewpoint for historical reasons that turns out to be sub-optimal?

It turns out that there is at least one.

You can read it up here.

All the circle formulae use radii, but pi is defines as circumference by diameter. A full 360 in radians is 2 Pi. It makes a lot more sense using tau, where:

\tau = 2 \pi

So it turns out that all these centuries we have used the wrong constant. In Carl Sagan’s “Contact” aliens send the digits of Pi as the header to their messages, since this is a recognisable sequence that could not have arisen by natural processes. But they would have sent tau not pi.

Just like rotational numbers, the tau constant does not break existing laws or adds new ones. It is just a better and more natural interpretation of existing knowledge.

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