Two things I always watch for with supplements, even with guidance from my doctor: Does it have a CGMP certification, and is there any data to back up the marketing claims? CGMP stands for “Current Good Manufacturing Process,” which are FDA guidelines put in place for a product’s safety. This includes where and how it was made, as well as what it was made of. But even with this kind of baseline, it’s hard to tell what additives are used in a supplement and how that can counteract its effects or react with your body chemistry.

So, Can I Take Melatonin or Not?

I tell you these things out of an abundance of caution. If your doctor gives you the go-ahead to use melatonin, follow their advice. Kuhlmann says he advises his patients to start at 3 milligrams but to never take more than 10 milligrams. For kids, he also urges speaking to a pediatrician and/or a sleep medical professional.

Melatonin also can’t do all the heavy lifting, and timing is crucial. As part of maintaining good bedtime habits, he also emphasizes the importance of taking it on time, at the same time, nightly. This will help establish the wind-down routine your brain follows via its circadian rhythm; as we established earlier, melatonin is supposed to lead this process.

If you’re keen on keeping melatonin in your bedtime routine, WIRED reviewer Molly Higgins tested and recommends Onnit’s Instant Melatonin Spray, which comes in two flavors: mint and lavender. (She tested the latter.) The standard serving size—six oral sprays—equals 3 milligrams of melatonin, which she found immediately made her sleepy. She did find she needed to increase her dosage over time to attain the same result, but, as we established above, it’s best to stay within the 10 milligram threshold.

For those of you who are just researching options, consider these alternatives. Diet and exercise, as tired as you may be of hearing that, really are essential to getting good, quality sleep. Case in point: You took a CrossFit class for the first time and are ready to conk out right after you manage to get dinner and a shower. Something else to consider: Perhaps relying on melatonin or a sleep supplement is treating a symptom you’re dealing with, as opposed to the actual issue impacting your sleep.

Bedtime Habits

Photograph: Martin Cizmar

LectroFan

High Fidelity White Noise Machine

It’s hard to overstate the importance of sleep hygiene and bedtime routine habits—a sleep supplement won’t be the end-all and be-all. Also, no caffeine after a certain time of day—you know your body best, but I’d say early afternoon at the very latest. Also, I’m sorry to my fellow readers and late-night scrollers, but devices need to be put away an hour or more before bedtime. Blue light that radiates from devices’ screens mimics that of sunlight, and your brain can’t discern the difference. All it knows is there’s still “daylight” that you need to be awake for, and that prolongs the falling-asleep process.

Other alternatives to supplements in the pursuit of better sleep can include sound machines (my favorite is above), where various frequencies of noise lull you to sleep. We’ve also tested sleep gadgets pretty extensively to not only get us to sleep, but also maintain deep sleep.

And maybe melatonin is a different sort of band-aid over the reality of your sleep situation, meaning that your mattress may need to be replaced. We have plenty of mattresses we’ve tested for every kind of sleeper, along with the best sheets and pillows. Supplements may not be the answer, after all, but more of a sleep space upgrade—all things to consider!

Compare Our Picks

Others Tested

Sakara Life; starts at $141 per week; up to $465 for specialty programs: This plant-based, gluten-free meal kit reminds me of what most people think when they think of “crunchy” vegan food—raw vegetables with an earthy taste. Nearly all meals in Sakara’s lineup are uncooked and preprepared—items like veggie burgers are without buns, lasagnas are “deconstructed.” For example, a “Lavender Quesadilla” has broccoli pesto and cashew “cheese” with hibiscus salsa … you get the idea. The menu is curated each week, and meals come in single servings. Sakara also has health supplements (which can be scientifically dubious), like a metabolism booster and fulvic acid cell reset. Sakara’s signature nutrition program meal plan is designed to replace all meals and is delivered twice weekly. If you buy one week of five days, three meals a day, it’s $465 per week; weekly subscriptions of five days, three meals a day, is $395 per week; prices go down to $141 per week with a 12-week subscription for three days at two meals per day. There’s also a “Level II: Detox” program, starting at $465 per week. This meal kit seems fit for Gwyneth Paltrow or WAGs (wife or girlfriend of professional athletes) everywhere, but it wasn’t the right fit for my budget and taste preferences.

NutriFit

NutriFit for $10 to $45 per meal: NutriFit is more like a personal chef than a meal-kit delivery service, specializing in nutrient-dense, fully prepared meals with a huge range of fare, with gluten- and dairy-free and vegetarian and vegan options. The company ships to the lower 48 states, and most meals hovered around $20. NutriFit has customized, chef-curated meal plans that are tailored for the eater and include specifics like health goals and dietary restrictions, where the customer can select their own meals on the Premium plan or have the curated meals from the 13-week rotating menu, starting at $19 per day. There are also à la carte options, which I tested, which range from $10 to $45 per meal. These don’t require a subscription or a minimum, and come in meals that serve three to four people or in individual size Fit for ONE meals that feed one, where you choose from “Always Available Favorites” and rotating new specials. A lentil chickpea salad, cold udon noodles, hearty roasted tomato soup, and crispy vegan tacos were standouts. But I wasn’t a huge fan of most of the chef-curated specials, and the food started to wilt or get mushy if not eaten within the first few days. The user interface of the service isn’t the best or easiest to navigate, either.

Photograph: Molly Higgins

Fresh! Meal Plan from $11 to $14 per meal: You can choose from 6, 10, or 14 meals per week, or order à la carte (which is a minimum of eight meals), ranging from $11 to $14 per meal, with the price lowering the more you order. It’s got choices for keto, paleo, high-protein, dairy- and gluten-free, and vegan and vegetarian meals, and everything is preprepared and just needs to be microwaved (or air fried) for about three minutes. There were six vegan meals and four vegetarian meals at the time of writing, with a menu filter to easily see choices. The vegetarian coconut chia breakfast pudding and margherita breakfast pizza were standouts, the vegan crab cakes had a mushy consistency and almost cinnamon-like flavor, and the vegan blackened “chickn” and Cajun pasta was rubbery and lacked spice. Since testing several months ago, none of the plant-based meal choices has changed, so this may be best as a supplemental meal kit for plant-based eaters.

Not Recommended

Photograph: Molly Higgins

Eat Clean for $9 to $13 per meal: This vegan meal delivery service would be best for someone who loves the taste and convenience of TV dinners. Eat Clean has a dozen plant-based heat-’n’-eat meals available, with availability to order six to 20 meals per week, ranging from six meals for $13 each to 20 meals at $9 each. Each meal comes in a plastic container and needs to be microwaved or heated for around three minutes. Many of the meals have very similar flavors—the tomato sauce base for the chili, spaghetti, and lasagna all tasted the same. The meals with sides often felt random: zucchini with mac and cheese and nuggets; a cornbread on the side of chili that tasted exactly like a cinnamon coffee cake (the flavors didn’t go well together on that one). Like TV dinners, flavors were often one-note, and I opted to air fry to enhance mushy textures. This meal kit is nearly the same price as most I’ve tested, and the picks above are a whole lot tastier.

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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.