Why NASA Only Needs Pi To So Many Decimal Places

December 17, 2024 by Lewin Day 51 Comments

If you’re new to the world of circular math, you might be content with referring to pi as 3.14. If you’re getting a little more busy with geometry, science, or engineering, you might have tacked on a few extra decimal places in your usual calculations. But what about the big dogs? How many decimal places do NASA use?

*NASA doesn’t need this many digits. It’s likely you don’t either. Image credits: NASA/JPL-Caltech*

Thankfully, the US space agency has been kind enough to answer that question. For the highest precision calculations, which are used for interplanetary navigation, NASA uses 3.141592653589793 — that’s fifteen decimal places.

The reason why is quite simple, going into any greater precision is unnecessary. The article demonstrates this by calculating the circumference of a circle with a radius equal to the distance between Earth and our most distant spacecraft, Voyager 1. Using the formula C=2pir with fifteen decimal places of pi, you’d only be off on the true circumference of the circle by a centimeter or so. On solar scales, there’s no need to go further.

Ultimately, though, you can calculate pi to a much greater precision. We’ve seen it done to 10 trillion digits, an effort which flirts with the latest Marvel movies for the title of pure irrelevance. If you’ve done it better or faster, don’t hesitate to let us know!

Square Roots 1800s Style — No, The Other 1800s

November 25, 2024 by Al Williams 44 Comments

[MindYourDecisions] presents a Babylonian tablet dating back to around 1800 BC that shows that the hypotenuse of a unit square is the square root of two or 1.41421. How did they know that? We don’t know for sure how they computed it, but experts think it is the same as the ancient Greek method written down by Hero. It is a specialized form of the Newton method. You can follow along and learn how it works in the video below.

The method is simple. You guess the answer first, then you compute the difference and use that to adjust your estimate. You keep repeating the process until the error becomes small enough for your purposes.

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3D Space Can Be Tiled With Corner-free Shapes

November 23, 2024 by Donald Papp 28 Comments

Tiling a space with a repeated pattern that has no gaps or overlaps (a structure known as a tessellation) is what led mathematician [Gábor Domokos] to ponder a question: how few corners can a shape have and still fully tile a space? In a 2D the answer is two, and a 3D space can be tiled in shapes that have no corners at all, called soft cells.

These shapes can be made in a few different ways, and some are shown here. While they may have sharp edges there are no corners, or points where two or more line segments meet. Shapes capable of tiling a 2D space need a minimum of two corners, but in 3D the rules are different.

A great example of a natural soft cell is found in the chambers of a nautilus shell, but this turned out to be far from obvious. A cross-section of a nautilus shell shows a cell structure with obvious corners, but it turns out that’s just an artifact of looking at a 2D slice. When viewed in full 3D — which the team could do thanks to a micro CT scan available online — there are no visible corners in the structure. Once they knew what to look for, it was clear that soft cells are present in a variety of natural forms in our world.

[Domokos] not only seeks a better mathematical understanding of these shapes that seem common in our natural world but also wonders how they might relate to aperiodicity, or the ability of a shape to tile a space without making a repeating pattern. Penrose Tiles are probably the most common example.

That Coin Toss Isn’t Actually 50/50

October 25, 2023 by Donald Papp 19 Comments

A coin flip is considered by many to be the perfect 50/50 random event, even though — being an event subject to Newtonian physics — the results are in fact anything but random. But that’s okay, because what we really want when we flip a coin is an unpredictable but fair outcome. But what if that’s not actually what happens?

There’s new research claiming that coin tosses demonstrate a slight but measurable bias toward landing on the same side they started. At least, this is true of coin flips done in a particular (but common) way. Coins flipped with the thumb and caught in the hand land with the same side facing up 50.8 percent of the time.

The new research builds on earlier work proposing that because of human anatomy, when a human flips a coin with their thumb, the motion introduces a slight off-axis tilt that biases the results. Some people do it less (biasing the results less) and some do it more, but while the impact is small it is measurable. As long as the coin is caught in the hand, anyway. Allowing the coin to fall on surfaces introduces outside variables.

Therefore, one can gain a slight advantage in coin flips by looking at which side is facing up, and calling that same side. Remember that the flipping method used must be that of flipping the coin with the thumb, and catching it with the hand. The type of coin does not matter.

Does this mean a coin flip isn’t fair? Not really. Just allow the coin to fall on a surface instead of catching it in the hand, or simply conceal which side is “up” when the coin is called. It’s one more thing that invites us all to ask just how random is random, anyway?

Your Engineering Pad In Browser

May 29, 2023 by Al Williams 15 Comments

It was always easy to spot engineering students in college. They had slide rules on their belts (later, calculators) and wrote everything on engineering pads. These were usually a light brown or green and had a light grid on one side, ready to let you sketch a diagram or a math function. These days, you tend to sketch math functions on the computer and there are plenty of people willing to take your money for the software. But if you fire up your browser, head over to EngineeringPaper.xyz and you might save a little cash.

Although it looks a lot like a Jupyter notebook, the math cells in EngineeringPaper keeps track of units for you and allows you to query results easily. Want to read more? Luckily, there is an EngineeringPaper worksheet that explains how to use it. If you prefer your explanations in video form, check out their channel, including the video that appears below.

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Math Reveals How Many Shuffles Randomizes A Deck

May 28, 2023 by Donald Papp 52 Comments

Math — and some clever simulations — have revealed how many shuffles are required to randomize a deck of 52 cards, but there’s a bit more to it than that. There are different shuffling methods, and dealing methods can matter, too. [Jason Fulman] and [Persi Diaconis] are behind the research that will be detailed in an upcoming book, The Mathematics of Shuffling Cards, but the main points are easy to cover.

A riffle shuffle (pictured above) requires seven shuffles to randomize a 52-card deck. Laying cards face-down on a table and mixing them by pushing them around (a technique researchers dubbed “smooshing”) requires 30 to 60 seconds to randomize the cards. An overhand shuffle — taking sections from a deck and moving them to new positions — is a staggeringly poor method of randomizing, requiring some 10,000-11,000 iterations.

The method of dealing cards can matter as well. Back-and-forth dealing (alternating directions while dealing, such as pattern A, B, C, C, B, A) yields improved randomness compared to the more common cyclic dealing (dealing to positions in a circular repeating pattern A, B, C, A, B, C). It’s interesting to see different dealing methods shown to have an effect on randomness.

This brings up a good point: there is not really any such a thing as “more” random. A deck of cards is either randomized, or it isn’t. If even two cards have remained in the same relative positions (next to one another, for example) after shuffling, then a deck has not yet been randomized. Similarly, if seven proper riffle shuffles are sufficient to randomize a 52-card deck, there is not really any point in doing eight or nine (or more) because there isn’t any such thing as “more” random.

You can watch these different methods demonstrated in the video embedded just under the page break. Now we know there’s no need for a complicated Rube Goldberg-style shuffling solution just to randomize a deck of cards (well, no mathematical reason for one, anyway.)

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Native Alaskan Language Reshapes Mathematics

April 18, 2023 by Bryan Cockfield 83 Comments

The languages we speak influence the way that we see the world, in ways most of us may never recognize. For example, researchers report seeing higher savings rates among people whose native language has limited capacity for a future tense, and one Aboriginal Australian language requires precise knowledge of cardinal directions in order to speak at all. And one Alaskan Inuit language called Iñupiaq is using its inherent visual nature to reshape the way children learn and use mathematics, among other things.

Arabic numerals are widespread and near universal in the modern world, but except perhaps for the number “1”, are simply symbols representing ideas. They require users to understand these quantities before being able to engage with the underlying mathematical structure of this base-10 system. But not only are there other bases, but other ways of writing numbers. In the case of the Iñupiaq language, which is a base-20 system, the characters for the numbers are expressed in a way in which information about the numbers themselves can be extracted from their visual representation.

This leads to some surprising consequences, largely that certain operations like addition and subtraction and even long division can be strikingly easy to do since the visual nature of the characters makes it obvious what each answer should be. Often the operations can be seen as being done to the characters themselves, instead of in the Arabic system where the idea of each number must be known before it can be manipulated in this way.

This project was originally started as a way to make sure that the Iñupiaq language and culture wasn’t completely lost after centuries of efforts to eradicate it and other native North American cultures. But now it may eventually get its own set of Unicode characters, meaning that it could easily be printed in textbooks and used in computer programming, opening up a lot of doors not only for native speakers of the language but for those looking to utilize its unique characteristics to help students understand mathematics rather than just learn it.