Geometry Insights FAQ with Tiago Hands — Mathematics, Premium Content, Q.E.D. Engine & Function Atlas
Inside Geometry Insights with Tiago Hands
Tiago Hands discusses the philosophy behind Geometry Insights, why understanding mathematics from first principles matters, who the platform is designed for, and his thoughts on proof, education, artificial intelligence and the future of mathematical discovery.
What is Geometry Insights?
Geometry Insights is a mathematics platform I created to help students and independent learners make faster, deeper progress in pure mathematics.
The central idea behind the site is simple: I do not just want to show the final result — I want to show how you get there.
Mathematical ideas are broken down carefully and then built back up from first principles. I show the intermediate workings, derivations and reasoning in detail so that readers can understand how each step follows from the previous one, rather than being expected to fill in important gaps for themselves.
Geometry Insights covers areas including geometry, matrices, vectors, the dot product, the cross product, functions, parametrisations, proofs and other topics across pure mathematics.
Who is Geometry Insights mainly aimed at?
Geometry Insights is aimed at a broad range of people who want to strengthen their mathematical understanding.
It is particularly useful for GCSE, A-Level and A-Level Further Mathematics students who want to go beyond memorising methods and develop a stronger understanding of the mathematics behind them.
It is also aimed at undergraduate students who want to revisit material they may have forgotten, strengthen areas they feel less confident in, or fill gaps in their mathematical understanding.
Because the material is built carefully from first principles, it can be particularly useful when returning to topics such as matrices, vectors, functions and parametrisations that underpin more advanced study.
The site is also suitable for adult learners and independent learners who want to return to mathematics or explore it in greater depth outside formal education.
Another important audience is people interested in computer graphics, mathematical art and mathematical visualisation. Topics involving vectors, matrices, transformations, parametrisations and functions naturally connect pure mathematics with graphics and visual creativity.
Why do you like proof so much and deriving equations from first principles?
Because I want to understand the formulas we use, not simply memorise them and plug numbers into them.
For me, one of the most rewarding parts of mathematics is reaching that eureka moment when you suddenly understand where an equation comes from, why it works and how different mathematical ideas connect.
Once that happens, mathematics stops feeling like a collection of isolated formulas and starts to reveal a much larger structure.
That curiosity has always been important to me. I never liked asking how a formula could be derived and effectively being told that I did not need to know — that I should just learn the equation, use it in the exam and move on.
I always wanted to know why.
I think telling somebody to use a formula without helping them understand where it comes from misses one of the most valuable parts of mathematics. It may teach someone how to complete a calculation, but it does not necessarily help them understand the mathematics behind it.
That is one of the reasons proof and derivation are so central to Geometry Insights. I like taking equations apart, tracing them back to their foundations and then building them up again carefully.
When you understand where a formula comes from, you no longer have to treat it as something mysterious that has simply been handed to you.
What makes Geometry Insights special?
I think one of the things that makes Geometry Insights special is the level of detail.
The content is designed to help fast-track progress in further pure mathematics by breaking ideas down and then building them back up from first principles.
Subscribers can strengthen their understanding of topics such as matrices, vectors, the dot product, the cross product, functions and mathematical terminology, while also exploring a growing collection of proofs and derivations.
I want people to be able to see exactly how one line follows from the previous one.
For me, the journey through the mathematics is just as important as the final result.
Presentation also matters. Mathematics should not only be correct; it should be clear, readable, well structured and enjoyable to explore.
Geometry Insights is continually evolving as well. I keep adding new mathematics, improving existing material and developing new tools and resources. I want the platform to become better and more useful as the library grows.
What are your favourite features on the website?
One of my favourite features is Copy Proof in UTF-8.
It allows subscribers to copy premium mathematical proofs and workings in a clean, human-readable UTF-8 format and paste them wherever they need them.
I particularly like this because the mathematics is not trapped inside the visual presentation of the webpage. Subscribers can take the workings with them and use them in notes, documents or elsewhere while preserving clear mathematical notation.
I also really like the idea of “proof on demand.”
That is one of the ideas behind the evolving Q.E.D. Engine. As the collection of mathematical material grows, I want people to be able to search for a proof, theorem or mathematical result and reach the relevant material quickly.
Detailed mathematical explanation, derivation, proof and worked reasoning.
geometryinsights.com →An evolving approach to finding proofs and mathematical results quickly — “proof on demand”.
qedengine.com →Visual and interactive exploration of functions, graphs and their mathematical behaviour.
functionatlas.com →Why should people subscribe to Geometry Insights?
Before creating Geometry Insights, I spent years sharing free mathematical proofs and workings online, and I received a very positive response to that work.
I have built an audience of more than 66,500 followers on Instagram, over 8,400 YouTube subscribers and more than 35,000 followers on Threads.
I think people respond to the way I present mathematics. I like to show the complete thought process, make the working easy to follow and avoid skipping the steps that can be the difference between simply seeing a result and genuinely understanding it.
Geometry Insights takes that approach much further.
Subscribers currently have access to 196 premium articles and counting, together with 18 premium resources.
The premium format gives me the space to develop mathematical ideas properly, show complete derivations, explore related concepts, include interactive material and continually improve the way the mathematics is presented.
Social media is excellent for introducing an idea or sharing a proof, but Geometry Insights allows me to go considerably deeper.
What do you think about the advancement of AI in mathematics?
I think the advancement of AI in mathematics has the potential to be very good for humanity.
Used well, I hope it helps create more critical thinkers, engineers, physicists, mathematicians, doctors and scientists of all kinds — people who can use deeper knowledge to help humanity progress both intellectually and from a health standpoint.
One of the things that excites me most is the possibility of making high-quality education and mathematical support far more accessible.
I think AI could be particularly valuable for disabled people, people from lower-income backgrounds and people with learning disabilities or additional learning needs.
If these technologies can give more people access to clear explanations, patient guidance and high-quality educational material, I think AI has the potential to become a powerful equalising force.
At the same time, I would not like to see AI used for killing, warfare or causing harm. I think that would be a dreadful use of such powerful technology.
There is an important distinction, though, between using AI to create harm and using it to protect people from harm.
I can see enormous value in AI being used for cybersecurity, protecting critical infrastructure, identifying serious threats and helping defend society against nuclear risks and other dangerous actions by bad actors.
If we can use this extraordinary capability responsibly — for education, science, medicine, security and genuine human progress — I think humanity could make some very important advances in the future.
I am especially excited about what may happen as advances in AI develop alongside mathematics, physics and quantum computing.
There is still so much that we do not understand about the universe.
I want to learn more about its structure, the mathematics that governs it, how everything connects and, ultimately, some of the deepest questions we can ask.
That sense of discovery is one of the things I find most exciting about the future.
Where can people find more information about Geometry Insights?
The main place to start is Geometry Insights.
That is the main home of the project and where people can explore the growing collection of mathematical articles, premium content, detailed proofs, derivations, resources and new features.
Those social platforms are where many people first discover my mathematical work, but Geometry Insights is where I have the space to develop the mathematics in much greater depth.
