AI Masters Olympiad Geometry
The team behind Google’s Deep Mind have just released details of a new AI system: AlphaGeometry This has been specifically trained to solve classical geometry problems – and already is now at the level of a Gold Medalist at the International Olympiad (considering only geometry problems). This is an incredible achievement – as in order to solve classical geometry problems, an AI system has to process both visual data and text data, understand the end goal, consider which additional constructions are required and then form a logical chain for a valid proof.
Simple geometric proof
Image from here.
A simple example of AlphaGeometry solving a geometry problem is shown at the top of the page. We are given an isosceles triangle and asked to prove that angles ABC and BCA are equal. To do this it draws an additional construction AD such that we have the 3 sides of the triangle ABD and ADC the same. As we have 2 congruent triangles with Side-Side-Side, angles ABC and BCA are equal.
Olympiad level mathematics
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The image above is from the 2015 International Olympiad. The Olympiad brings together 17-18 year olds competing in teams of six. To be chosen to even go to an Olympiad means you have to be one of the best young mathematicians in your country and even for these students this problem would be seriously challenging.
Olympiad level proof
Image from here.
AlphaGeometry was able to solve this problem – with an incredible 110 step proof spanning 8 pages. A visualisation of the extra constructions (in blue) needed for this proof are shown above. You can see the full proof here.
How good is AlphaGeometry?
Image from here.
We can see from the bar chart above that AlphaGeometry is a huge advance on the previous AI systems. The current model was able to get an average of 25/30 geometry problems correct – which would take it to around the level of a Gold Medalist at the International Olympiad. Possibly even more incredible is the fact that AlphaGeometry was not trained on Olympiad problems – or any geometry problems. Instead it started by generating 1 billion random geometrical diagrams and then derived all relationships between points and lines contained within them. This method allows AlphaGeometry to discovers theorems and relationships for itself – without the need to be told them in its programming.
What is the future of AI in mathematics?
Our long-term goal remains to build AI systems that can generalize across mathematical fields, developing the sophisticated problem-solving and reasoning that general AI systems will depend on, all the while extending the frontiers of human knowledge.
The team behind AlphaGeometry provide the above quote about their long term ambitions – the next step is to create a single AI system that is able to draw on expertise across multiple mathematical fields and then apply reasoning to create new proofs and solve problems in mathematics that humans have been unable to. Given the deep connections between different fields of mathematics, creating an AI expert across multiple fields could unlock many mathematical mysteries.
You can read the full research paper about AlphaGeometry here.