The original version of this story appeared in Quanta Magazine.

Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.

The world is full of such shapes—ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.

This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces—leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis, and physics.

Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”

So what are manifolds, and what kind of vocabulary do they provide?

Ideas Taking Shape

For millennia, geometry meant the study of objects in Euclidean space, the flat space we see around us. “Until the 1800s, ‘space’ meant ‘physical space,’” said José Ferreirós, a philosopher of science at the University of Seville in Spain—the analogue of a line in one dimension, or a flat plane in two dimensions.

In Euclidean space, things behave as expected: The shortest distance between any two points is a straight line. A triangle’s angles add up to 180 degrees. The tools of calculus are reliable and well defined.

But by the early 19th century, some mathematicians had started exploring other kinds of geometric spaces—ones that aren’t flat but rather curved like a sphere or saddle. In these spaces, parallel lines might eventually intersect. A triangle’s angles might add up to more or less than 180 degrees. And doing calculus can become a lot less straightforward.

The mathematical community struggled to accept (or even understand) this shift in geometric thinking.

But some mathematicians wanted to push these ideas even further. One of them was Bernhard Riemann, a shy young man who had originally planned to study theology—his father was a pastor—before being drawn to mathematics. In 1849, he decided to pursue his doctorate under the tutelage of Carl Friedrich Gauss, who had been studying the intrinsic properties of curves and surfaces, independent of the space surrounding them.

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For more inspiration, check out some of our recently updated buying guides, including the Best Office Chairs, the Best Cheap Phones, and the Best Space Heaters.

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Anker Laptop Power Bank for $88 ($47 off)

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There was an on-page coupon (PIXEL10) that had the best price we’ve tracked for any of the phones in the Google Pixel 10 lineup. That coupon is not available as of Saturday morning, but it may be back—clip it if you see it. This is still a good deal on the smartest Android phones you can buy, with fantastic cameras, snappy processors, gorgeous displays, and more AI integration than the average person needs. Check out our dedicated buying guide to figure out which Google Pixel 10 is right for you. If you’re in the market for an upgrade, now is a good time to buy considering that we’ve never seen any phone in this flagship lineup sell for less.

Bruvi BV-01 Brewer Bundle for $228 ($120 off)—Clip the Coupon

Photograph: Louryn Strampe

I’ve tested a lot of pod coffee makers, and the Bruvi BV-01 is my favorite. This deal price is the best we see outside of special events like Black Friday and Cyber Monday. The brewer is cute and looks great on a counter, with a large reservoir, an intuitive touchscreen display, and a built-in wastebin that collects used pods for you. The best part are the proprietary B-Pods, which are designed to biodegrade in a landfill. The bundle gets you the machine plus an assortment of bestselling coffee and espresso pods to get you started.

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Hydro Flask Standard Mouth Water Bottle for $30 ($10 off)

Photograph: Dick’s Sporting Goods 

Hydro Flask

Standard Mouth Water Bottle

This budget-friendly deal gets you a steal on the best reusable water bottle. Hydro Flask bottles are durable, portable, and easy to cover in all the stickers you’ve been hoarding. The handle is flexible, the bottle is leakproof, and every component is dishwasher safe (though you may want to opt for hand-washing if you do end up plastering it in stickers). A few different colors are on sale at this price.

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The most immediately striking difference is that Hyperkin’s product swaps the typical Xbox approach of asymmetric thumbsticks for the PlayStation’s horizontal layout. It also separates the D-pad (it’s one piece inside the pad, but splits its cardinal directions so each appears to be its own button), while the ABXY face buttons are spaced slightly further apart. Where the DualSense’s touchpad would sit, we have the Xbox home, menu, view, and share buttons, all blended in rather smartly. An LED ring around the home button just about echoes the lights running the periphery of the DualSense’s touchpad, although it’s really more of an inversion of the regular Xbox controller, where the home button itself lights up.

The Competitor’s thumbsticks come equipped with thumbcaps that mirror the PS5’s, an outer ring with a convex central point, but a pair of Xbox-standard concave caps are included. These easily pop on and off, and can be mixed and matched, if you were so (strangely) inclined.

There are two areas where this departs from both the standard Xbox and PlayStation controllers in terms of inputs. The first is the presence of two programmable rear buttons, M1 and M2. By default, these duplicate the input of the A and B buttons, but holding down the Mode button between them lets you remap them. There are also physical button locks to prevent their use entirely. The other is that while the Competitor boasts a 3.5-mm headphone jack like Microsoft’s official pad, it adds a built-in audio mute button, hidden in the black between the thumbsticks—a nice little upgrade.

Oddly Familiar

In use, the Competitor feels … well, a lot like a PS5 pad. The slightly wider grip fits in the hand comfortably, all inputs are accessible, and those symmetrical thumbsticks sit nicely in reach for all but the smallest hands. A microtextured underside provides a solid grip that, when coupled with its 232-gram weight, makes the Competitor feel particularly suited to longer play periods. It’s all very familiar if you’re already a multiformat gamer, to the extent that it sometimes slightly threw my muscle memory off, reaching a thumb out to do a PlayStation touchpad function and finding only the Xbox system buttons.