Designing Engare, a game born of a charming geometry problem

“When I was in high school, one day my geometry teacher asked a question and everyone in the class was so fascinated by that question -- even those students who were not so interested in math or geometry," says Mahdi Bahrami, developer of Engare. 

"The teacher asked the class what shape would be traced by a point attached to a ball if the ball was rolled across a surface. That day I decided to make a little game based on this question.”

Engare is a puzzle game about the beauty of mathematics and of Middle Eastern artwork, exploring the rules that explain the movements of an object in motion and the striking shapes that get created by these movements.

Players place a dot on a moving shape in hopes that the dot’s movement will replicate a specific design, watching the object move to know where to place their mark to recreate it. Soon, players will be manipulating the shape as well, knowing what shape, and where to place their dot, to draw what they need to complete a puzzle.

Through tracing lines with moving objects, Engare aims to show players the evocative power of a repetitive motion, having them create their own evocative art through where they place their dot on the shapes. It is a puzzle game, but one focused on beauty as well as brainteasing, helping teach the player to be curious about what these motions can create, and growing a desire within them to explore what emotions and art they too can build with it.

Finding inspiration in mathematical motion  

Unlike many puzzle games, the challenges were not entirely created within the developer’s mind. For Bahrami, finding inspiration in Middle Eastern artwork and in mathematical rules was key, giving him ideas for movements and rules he could place into Engare to create striking things the players could witness come together. 

“I had made few puzzle games before Engare. For designing levels for those games, the main question was always: ‘How can I make a more challenging/difficult puzzle’," he says. "In the case of Engare, it was a bit different. It was not about designing difficult puzzles. The main challenge was finding interesting ideas which could be expressed through the drawing system. These puzzles could be very simple and easy, but had interesting and unexpected results.”

Bahrami’s desire to see the beauty of motion out in the world and transfer it into his game’s drawing system took him to many different places as the developer sought different types of motion, mathematical rules given form, and the ways in which it was reflected in Middle Eastern artwork. 

“The main challenge was finding interesting ideas for each level. I couldn’t just sit in front of my computer and design new levels. I needed to find inspiration from somewhere outside of my video game," Bahrami says. "For example, I designed a handful of levels after I visited the Mathematics Exhibition of the London Science Museum, and got inspired by a Geometric Chuck device created by John Jacob Holtzapffel.”

“The last two sections of the game were highly inspired by this device. It is an arrangement of mechanism for producing two or more circular movements in parallel planes. The combination of these movements with different velocity ratios and different radiuses results in the formation of interesting curves and geometric figures,” he continues.

Bahrami could see the stirring shapes that this mechanism could create. It’s a machine carrying through a very specific motion, but when that motion is traced, or altered by making tiny adjustments to radiuses and speeds, it creates unique new visual patterns for the player to explore. To the naked eye, it is a machine doing its job, but when one follows the line of its motion, there is an art there – a beauty in motion that Bahrami wanted to capture in Engare.

Other mechanisms would be worked into the game as well. “I was looking for different type of motions which were mathematically interesting. An example for this type of ideas could be Tousi Couple, a mathematical device discovered by the Persian astronomer and mathematician Nasir Al-Din al-Tusi,” says Bahrami.

“The basic idea behind Tousi Couple is that a small circle rotates inside a larger circle twice the diameter of the smaller circle. If you draw the trajectory of a point on the circumference of the smaller circle," he adds. "The result will be a line segment. This is interesting because normally you don’t expect a rotating circle to draw a straight line."

In experiencing these mathematical machines in motion, Bahrami learned what he wanted to place inside Engare to excite the player’s curiosity, getting them to see what striking imagery they could create from their own experiments with the objects in motion – what things they could tease from these rules given form.

Making a game about finding beauty in lines

This kind of beauty is something people had been exploring for a very, very long time already. “I realized that these mathematical motions have a strong connection with abstract art, something especially seen in the Middle Eastern art and architecture,” says Bahrami.

Engare has some powerful connections to Middle Eastern artwork within its design, drawing from very similar places from its history

“Ancient artists in the Middle East were not allowed to draw living creatures. They had to avoid figurative representations because it was considered as a sin. Instead, they had to use abstract geometrical shapes to express their feelings," says Bahrami. "So, a large portion of art and architecture in the Middle East became inspired by mathematics. Those shapes drawn in Engare have the same mathematical rules behind them.”

Bahrami looked to teach players to use a similar kind of expression through Engare. In giving them mathematical rules in motion, he wanted to grant the player access to an easy-to-use tool that would let them express themselves through this same art style, showing them that they could also create evocative, personal imagery through what appear to be rigid rules. 

“I think we are naturally curious about seeing where an object in motion takes us - especially when we recognize the patterns of motion," says Bahrami. "Maybe we don’t know the mathematical equations behind the motions, but when it’s visualized, we understand the patterns and are therefore more curious to try different experiments with the system.”

It wasn’t necessary for the player to understand the rules that governed each motion in the beginning, but rather to spark that curiosity in the player. It was important for them to see what was possible within this structure that had so fascinated Bahrami. It was to tease the art and emotion and creativity out of the player through its motions.

The puzzles in Engare, therefore, are more to draw that kind of thought and feeling out of the player – to teach them what they can do with these objects in motion, and make them curious. “The puzzles help me to show the players what I find interesting about these mathematical rules. Then, the drawing tools give them the ability to explore and draw what they find interesting,” says Bahrami.

Engare is a puzzle game, and one that will challenge the player’s mind, to be sure. Engare’s goal is not just to keep the mind busy for a few hours, though, but to excite the imagination with the same possibilities that can be seen in striking Middle Eastern art and architecture. It’s an exploration of the beauty in mathematical thought, and to draw the player into seeing what they, too can create through its seemingly strict motions.