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.

Bode Plot Un-Lecture

March 8, 2023 by Al Williams 4 Comments

[Rolinychupetin] insists that his recent video is not a lecture but actually a “recitation” about Bode plots. That may be, but it is still worth a watch if you want to learn more about the topic. You can see the video below.

If you haven’t run into Bode plots before, they are simple plots of magnitude or phase vs. frequency, usually plotted on a log scale. Named after Bell Lab’s [Hendrik Wade Bode], they are useful for understanding filters or anything with a frequency response.

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Organic Fibonacci Clock Is All About The Spiral

January 12, 2023 by Lewin Day 5 Comments

Whether you’re a fan of compelling Tool songs, or merely appreciate mathematical beauty, you might be into the spirals defined by the Fibonacci sequence. [RuddK5] used the Fibonacci curve as the inspiration for this fun clock build.

The intention of the clock is not to display the exact time, but to give a more organic feel of time, via a rough representation of minutes and hours. A strip of addressable LEDs is charged with display duty. The description is vague, but it appears that the 24 LEDs light up over time to show the amount of the day that has already passed by. The LEDs are wound up in the shape of a Fibonacci spiral with the help of a 3D printed case, and is run via a Wemos D1 microcontroller board.

It’s a fun build, and one that we can imagine would scale beautifully into a larger wall-hanging clock design if so desired. It at once could display the time, without making it immediately obvious, gradually shifting the lighting display as the day goes on.

We’ve seen other clocks rely on the mathematics of Fibonacci before, too. If you’ve cooked up your own fun clock build, don’t hesitate to let us know!

a kinetic bar framework mounted on a wooden base made of 3d printed bars of alternative black and grey color, each joined with m3 bolts and nuts

Kinetic Cyclic Scissors

October 26, 2022 by Abe Connelly 6 Comments

[Henry Segerman] and [Kyle VanDeventer] merge math and mechanics to create a kinetic cyclic scissors sculpture out of 3D printed bars adjoined together with M3 bolts and nuts.

The kinetic sculpture can be thought of as a part of an infinite tiling of self similar quadrilaterals in the plane. The tiling of the plane by these self similar quadrilaterals can be realized as a framework by joining the diagonal points of each quadrilateral with bars. The basic question [Henry] and [Kyle] wanted to answer was under what conditions can the realized bar framework of a subsection of the tiling be made to move. Surprisingly, when the quadrilateral is a parallelogram, like in a scissor lift, or “cyclic”, when the endpoints lie on a circle, the bar framework can move. Tweaking the ratios of the middle lengths in a cyclic configuration leads to different types of rotational symmetry that can be achieved as the structure folds in on itself.

[Henry] and [Kyle] go into more detail in their Bridges Conference paper, with derivations and further discussions about the symmetry induced by adjusting the constraints. The details are light on the actual kinetic sculpture featured in the video but the bar framework was chosen to have a mirror type of symmetry with a motor attached to one of the central, lower bars to drive the movement of the sculpture.

The bar framework is available for download for anyone wanting to 3D print or laser cut their own. Bar frameworks are useful ideas and we’ve seen them used in art sculptures to strandbeests, so it’s great to see further explorations in this space.

Video after the break!

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