## Mathematics: the Joy of Intuition and Revelation

Ruebena Dawes, The University of Sydney

I didn’t begin my mathematics degree out of a particularly strong passion for the subject. I had always enjoyed maths, but it was largely a pragmatic and spontaneous choice, driven by a rousing and convincing speech at the open day of my university. Over the last three years, however, my love and awe for the subject has grown dramatically. My favourite part of my degree has been the moments of clarity wrought by stretching my mind to understand new mathematical concepts.

One simple example came at the tail end of a unit on ordinary differential equations. In leading us through the Poincare-Hopf Index Theorem, my lecturer asked us to consider the Cartesian plane (shown right) and asked us why we couldn’t see the axes intersecting at infinity like we could see them intersecting at 0 (the origin). He then asked us to imagine the plane as a sphere that also intersects at infinity, and said that the reason we can’t see it intersect is that we are viewing it from that point- we are looking at the origin from infinity (shown below). If we can change our perspective to the origin (by a change of coordinates), then we will be able to see the plane intersect at infinity.

There was more to the theorem, but I just found this concept so stimulating. Something as plain and simple as the x,y-axes could be thought of in a totally different way. I was so excited that I explained it to my mum. She found it interesting but was mostly puzzled. I remember her saying “I just didn’t know that that kind of thing was maths”.

I think many people experience this same confusion when they hear enthusiasts expound on the beauties of mathematics, and I don’t blame them. When I think back to learning and using the quadratic equation ad nauseam in high school I don’t feel particularly inspired. Mathematics is an exceedingly useful discipline. learning to apply theorems and crunch the numbers is a great life skill, but it is not all there is to appreciate about maths.

The example I gave above is not meant to serve as a profound example of the beauty of maths, it was just something I recall finding thought-provoking. But I do mean it to serve as an example of one of the foremost things I discovered throughout my degree: Learning mathematical concepts that require stretching your imagination and extending your mind, is as enriching to the soul as reading poetry or viewing a great piece of art. To me, the joy of mathematics is the joy of intuition and revelation, and I am so grateful for the fact that I have been able to study such a unique discipline these past three years. I am looking forward to spending the rest of my life learning more mathematics and seeking out more of these moments of intuition and revelation.

Ruebena Dawes was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.

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