Vantage

Inspiration

Geometry is often taught through formulas, diagrams, and static examples, but some of its most interesting ideas are difficult to understand until you can actually interact with them. Non-Euclidean and impossible geometries challenge many of the assumptions we naturally make about space: the closest object may not actually be the closest, a path that looks disconnected can become connected from another viewpoint, and a triangle does not always have angles that add up to 180°.

We wanted to turn these strange mathematical ideas into something that feels intuitive, playful, and discoverable.

That led us to Vantage: A word puzzle where perspective bends space and geometry changes the rules.

Inspired by Scrabble, Monument Valley, and perspective-based puzzle games, Vantage combines word building with impossible architecture and hyperbolic geometry. Instead of simply showing players unusual geometry, we wanted geometry itself to become part of how they think, explore, and solve puzzles.

Who It's For

Vantage is designed for anyone who wants to experience how fascinating and strange geometry can be without needing an advanced mathematics background.

By combining geometry with a familiar word-building game, we make mathematical ideas approachable through experimentation rather than equations alone. Players discover that changing their viewpoint can completely change how the board connects, while the Hyperbolic Chamber challenges their normal intuition about distance.

We especially see Vantage as an educational experience for younger players and students. The game encourages spatial reasoning, creativity, experimentation, and problem-solving while introducing mathematical concepts in a way that feels like playing rather than studying.

Our goal is simple: you should not need to understand non-Euclidean geometry before playing Vantage. Playing Vantage should help you begin to understand it.

How we built it

We built Vantage using Three.js for the 3D graphics and hyperbolic rendering, JavaScript, Vite, and the Web Audio API.

The main game takes place on an impossible monument made of interconnected blocks, floating arms, and a Penrose-inspired crown. Every face can contain letter tiles. Using an orthographic camera, structures that are physically separated in 3D can visually align from specific vantage points. When they align, their word paths become connected, allowing players to construct words across otherwise disconnected surfaces.

The game mathematically determines these special viewpoints by analyzing when two tile strips project onto the same location and direction on the screen.

We also built a Hyperbolic Chamber using the Poincaré disk model. Before playing words, players are shown crystals that appear to be at similar distances and must determine which one is truly closest. Because hyperbolic space expands dramatically toward the edge of the disk, visual distance can be misleading. After making a choice, the game reveals the actual hyperbolic geodesics and distances.

Finally, we created a Hyperbolic Lab where players can freely experiment with curved space: walking around the Poincaré disk, creating hyperbolic triangles, observing their angle sums, and seeing how familiar Euclidean behavior changes.

Challenges we ran into

One of our biggest challenges was making the geometry mathematically meaningful without making the game confusing.

For the impossible monument, simply making objects look connected was not enough. We needed to determine when separate surfaces actually align from a particular camera direction, ensure that the connection is visible rather than obstructed by another block, and make the interaction understandable to the player.

Hyperbolic geometry presented an entirely different challenge. Ordinary Euclidean distance and movement no longer work. We had to implement hyperbolic distance, geodesics, Möbius transformations, and movement inside the Poincaré disk while still making everything responsive enough to feel like a game.

Another major challenge was balancing education with gameplay. We did not want Vantage to become a mathematical demonstration with a game placed on top of it. The geometry needed to directly affect the player's decisions.

Accomplishments that we're proud of

We're especially proud that the non-Euclidean geometry in Vantage is not merely visual decoration — it changes how the game is played.

Players can form words across structures that are disconnected in physical 3D space but become connected from the correct viewpoint. The Penrose-inspired crown creates a word path that visually forms an impossible continuous loop. Players can even reveal the trick and rotate away from the special viewpoint to see the real gaps hidden by the projection.

We're also proud of our Hyperbolic Chamber and Lab. The player can experience properties of hyperbolic space directly, including geodesic distance, triangles whose angles sum to less than 180°, and a boundary that remains infinitely far away no matter how far they walk.

Bringing word games, impossible perspective, and mathematically accurate hyperbolic geometry together into one coherent experience is the part of Vantage we're most proud of.

What we learned

Building Vantage changed the way we thought about geometry.

We learned that changing the underlying geometry of a world affects much more than its appearance. Distance, movement, angles, adjacency, and even what it means for two objects to be "next to" each other can change.

Technically, we learned how to work with the Poincaré disk model, hyperbolic distance, geodesics, Möbius transformations, orthographic projection, visibility testing, and procedural rendering.

But our biggest lesson was about interaction design: complicated mathematical concepts can become much easier to understand when users are allowed to discover their consequences themselves.

Instead of telling someone that hyperbolic space behaves differently, we can let them choose the crystal that looks closest — and discover that their Euclidean intuition was wrong.