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On curvature: cars, clothoids and cartography

Rollercoasters and railways were originally created without using curvature calculations. People on them would experience undesirable jerk (a sharp change in curvature results in jerk or centripetal forces). You may still experience this on old railways. The image below depicts a centrifugal railway that was constructed in 1846. Nowadays, rollercoaster and railway designers use a type of curve called a clothoid or Euler spiral to make the change in curvature less abrupt, for example when riding a loop-the-loop. I will later mention a couple of applications of Euler spirals. So curvature is clearly an important concept. Let’s get to grips with how it works, and where we should consider it. Continue reading

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Book of the Year 2019

A few weeks ago, we announced the 9 book shortlist for the 2019 Chalkdust Book of the Year. In this post, we announce the winner of this award.

We will be awarding two prizes: the Chalkdust Book of the Year (as chosen by our editors), and the Chalkdust Readers’ Choice (as voted for by our readers).

Chalkdust Book of the Year 2019

We found picking a winner of this award really difficult, as all the books on the shortlist were deserving of the prize (obviously, as if not they wouldn’t have made the shortlist). The one thing that set the winner apart from the other books was its appeal to both mathematicians and non-mathematicians. And the winner is:

The Maths of Life and Death

Kit Yates

This book (Amazon UK, Waterstones), as its title suggests, is all about the maths of life and death. It looks at the maths related to law, biology, medicine, the spread of diseases, and many more areas. Kit takes you through these areas with a series of interesting and well-written stories.

You can read our full review of The Maths of Life and Death here.

Chalkdust Readers’ Choice 2019

As well as picking our favourite from the shortlist, we held a vote for our readers to pick their favourite. The runaway winner of this poll was:

A Compendium Of Mathematical Methods: A handbook for school teachers

Jo Morgan

This book (Amazon UK, Waterstones, Foyles) contains details of various methods of completing various mathematical tasks, including subtraction, multiplication, simplifying surds, and polynomial division. Each method is demonstrated by example and reasons why each method works are discussed. This book is an excellent reference for secondary teachers, as well as an interesting book to browse through for the mathematically curious.

You can read our full review of A Compendium Of Mathematical Method here.

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The Hidden Half

Michael Blastland is a writer and broadcaster, and was the creator of BBC Radio 4’s More or Less. His book The Hidden Half: How the World Conceals its Secrets (Amazon UK, Waterstones), looks at how to understand, and how to avoid misunderstanding statistics and research.

Style

This book is less of a maths book than many of the books on our shortlist, and much that is discussed relates to science in general. But, as More or Less fans might expect from its creator, an awful lot of its content is related to statistics and evidence; enough that we here at Chalkdust have deemed this a maths book.

In The Hidden Half, Michael Blastland takes the reader through statistical and numerical ideas that most often lead to confusion: relative and absolute risks, the difficulty of comparing now with the past, and the effect of hidden factors. He takes you through each idea with a series of stories illustrating the effect and importance of each idea.

Control

The book begins with a story about marmorkrebs: a species of crayfish that can reproduce asexually, leading to their offspring being clones. Even though both the nature (genes) and nurture (enviroment) of the offspring could be kept the same for each child, there were still surprising differences in the appearance, size, and even internal organs of these offspring. From the go, this story had me hooked, and was a great illustration of an extra hidden causal factor.

Damage

This book looks at problems of the misinterpretation of statistics from the point of view of a scientist or economist, giving the effects of this misinterpretation a very real feel.

Aggression

I really enjoyed this one, and would strongly recommend it to many of my friends, especially my more science-liking (and less maths-loving) friends who would find themselves enjoying this before really realising that the ideas they’re reading about are mathematical. This makes for a very enjoyable read for a mathematician who will not be familiar with the stories that Michael tells.

 

Now that we’ve review all the book’s on this year’s shortlist, you can vote for your favourite. The book with the highest number of votes will be crowned the 2019 Readers’ Choice. Voting closes at 5pm on Wednesday 26 February.

What is your favourite book on the 2019 Book of the Year shortlist?

  • A Compendium Of Mathematical Methods: A handbook for school teachers by Jo Morgan (48%, 37 Votes)
  • Geometry Puzzles in Felt Tip: A compilation of puzzles from 2018 by Catriona Shearer (18%, 14 Votes)
  • Humble Pi: A Comedy of Maths Errors by Matt Parker (16%, 12 Votes)
  • So You Think You’ve Got Problems? by Alex Bellos (5%, 4 Votes)
  • The Art of Logic by Eugenia Cheng (4%, 3 Votes)
  • The Hidden Half by Michael Blastland (3%, 2 Votes)
  • Maths on the Back of an Envelope by Rob Eastaway (3%, 2 Votes)
  • The Maths of Life and Death by Kit Yates (3%, 2 Votes)
  • Here Come the Numbers by Kyle D Evans and Hana Ayoob (1%, 1 Votes)

Total Voters: 77

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Maths on the Back of an Envelope

Rob Eastaway is a recreational mathematician, author, speaker, and maths event organiser. His latest book, Maths on the Back on an Envelope (Amazon UK, Waterstones, Maths Gear (signed)), is all about how to (roughly) calculate pretty much anything.

Style

Maths on the Back on an Envelope is an easy-going and enjoyable read. It takes the reader through some methods of approximation of numbers, while showing the reader why being able to perform such approximations is a useful skill. As well as suggesting the use of approximation to check that the correctness of answers, or to work out the approximate size of answer that should be expected, the book also makes the very good point that often an approximation is the best that we can do, and giving answers to a greater-than-appropriate level of accuracy can be misleading.

Control

The book explains the methods of approximation in a very understandable manner, and also justifies their use and appropriateness. Throughout, the user is challenged to try out their newly found calculation skills; the solutions to these challenges are given the back of the book, alongside some discussion of some ways people go about answering them.

Damage

The discussion of estimation and accuracy in this book is very strong, and is not something I’ve read much about elsewhere. The book sets itself apart from other books by encouraging you to actively engage in tasks related to the material being discussed, and you may find yourself finishing it with a far greater understanding of what everyday numbers mean.

Aggression

Although many of the methods of mental or on-paper calculation discussed in this book will be familiar to the keen mathematician, this book is still an enjoyable read, and even the most capable calculator can benefit from thinking about the issues surrounding accuracy in this book. I’d most strongly recommend this book to those looking to improve their skills with numbers, or to improve their understanding of numbers around them an in the news, but there’s still plenty to enjoy here for others.

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The Maths of Life and Death

Kit Yates is a senior lecturer in the Department of Mathematical Sciences at the University of Bath, who works in mathematical biology. His first book, The Maths of Life and Death (Amazon UK, Waterstones), is all about the maths of life and death.

Style

The Maths of Life and Death takes the reader on a tour of seven important areas of mathematics that underpin life and death. Many of the ideas discussed are related to medical and biological applications of maths and stats, and included stories about when misunderstanding or misinterpreting numbers could have (and sometimes have had) very grave consequences.

Control

The maths in this book is well explained, and the application to real life issues both makes the maths more accessible to the less mathematical reader, and makes the book more interesting to the knowledgable mathematician who may be familiar with much of the maths but will be less familiar with the practical contexts where it can be applied.

Damage

The strongest sections of the book are those in which Kit discusses medical statistics and disease modelling. As his own research is based near these areas, he is able to draw on his personal knowledge of these areas, while maintaining reader-friendly explanations of the maths involved. There are many books about biology and medicine aimed at the general reader, but the maths involved in these pursuits is rarely presented to such an audience: this is this book’s biggest strength, and it sets it apart from many other maths books.

Aggression

I would recommend this book to those interested in maths, and those interested in medicine or biology as they will get a lot out of reading about the maths underlying much of the work in these areas. I have a few people in mind who might be getting this one next Christmas…

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Humble Pi

Matt Parker is a stand-up mathematician, YouTuber and all-round maths lover. You can read more about him in our interview with him that appeared in issue 09. His latest book, Humble Pi (Amazon UK, Waterstones, Maths Gear (signed)), takes the reader through a range of mathematical topics by looking at past errors that have been made in these areas.

Style

Humble Pi is a very entertaining read. Writing about mistakes, and the consequences of mistakes, allows Matt to include a number of amusing, and sometimes slightly frightening, anecdotes. These mistakes are used as an “excuse” to explain the underlying maths in an informative and engaging way.

Control

The mathematical ideas contained in this book are very clearly explained, and each one is attached to a real life mistake, making the importance of each bit of maths immediately clear.

Damage

As this book is about mistakes, and mistakes that have easily noticeable consequences make the best stories, the maths in this book is mostly that related to science and engineering. This means that the book spends time on topics that many maths books pass by, such as the resonance of bridges and financial maths. I enjoyed the financial maths section in particular, as this is an area that I don’t usually read about or spend much time thinking about.

Although readers may be aware of some of the the famous mistakes, such as the wobbly Millennium Bridge or Pac-Man breaking on the 256th level, there are many great stories here that readers will not be familiar with; and readers may well not know the true mathematical reasons behind the more famous stores.

Aggression

Overall, Humble Pi is both fun and interesting, and I would highly recommend it. I have in fact already given it to two friends as a present, and might be gifting it to more people this year now that the cheaper paperback is out.

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So You Think You’ve Got Problems?

Alex Bellos is a writer and broadcaster, and is The Guardian’s puzzle columnist. His latest book, So You Think You’ve Got Problems? (Amazon UK, Waterstones, Guardian bookshop), is a collection of puzzles.

Style

So You Think You’ve Got Problems? contains 125 puzzles, plus a selection of shorter ‘teaser’ puzzles spread throughout the book. Between the puzzles, Alex tells us of the historical context of the puzzles, points out connections between puzzles, gives small hints, as well as discussing what makes a puzzles good.

Control

The book ends with answers to all the puzzles, including clear explanations of how these solutions can be found.

Damage

As anyone who follows his Guardian puzzle column will know, Alex has very good taste in puzzles. Unsurprisingly, therefore, this is a great collection of puzzles, and there is plenty in this book to keep you thinking for a long time. I do a lot of puzzles, and some of the puzzles in this book were similar to puzzles I’d seen before, but there were plenty that were not familiar to me and lots of new situations to think about.

Aggression

I’d recommend this book to anyone who like puzzles and can think of a few people who I could give this to next Christmas…

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Here Come the Numbers

Kyle D Evans is maths teacher and an award-winning maths communicator and musical comedian. Hana Ayoob is an illustrator, science communicator, and creative producer of science events.

Together, they are the creators of Here Come the Numbers (Amazon UK, Waterstones, Explaining Science Publishing), the fourth book on the Chalkdust Magazine Book of the Year 2019 shortlist.

Style

Here Come the Numbers is a rhyming picture book. Each page of Here Come the Numbers features rhymes about numbers alongside illustrations of the ideas involved. At first glance, it looks like a children’s book, and it starts as many children’s maths books start: with small positive integers. But unlike many children’s maths books, it doesn’t stop there.

Damage

Here Come the Numbers quickly goes beyond the topics contained in your usual learning-to-count book. It introduces the reader to prime numbers, with illustrations of rectangles of objects to show factorisations, before touching on Pythagorean triples, negative number, and even Fermat’s last theorem. It does this while maintaining the easy-to-read and child-friendly style.

In a few places in the book, the reader is encourage to think about something, for example which squares can be broken into two smaller squares. These prompts provide the reader with opportunities to investigate mathematical ideas themselves.

Control

The ideas in Here Come the Numbers are very clearly articulated, and the illustrations work really well alongside the text to make the ideas easy to visualise.

Aggression

I would strongly recommend this to parents of young children, and I am eagerly awaiting the day when my niece is old enough to read it with her.

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The Art of Logic

Eugenia Cheng is a mathematician who is currently Scientist-in-Residence at the School of the Art Institute of Chicago. You can read more about her in our interview with her that was printed in issue 08. The third book on the Chalkdust Magazine Book of the Year 2019 shortlist is The Art of Logic: How to Make Sense of a World that Doesn’t (Amazon UK, Waterstones), Eugenia’s effort to drag formal logic down from its ivory tower and modern political discourse up from the gutter, and sit them round a conference table for a long overdue reconciliation.

Style, control, damage and aggression

Eugenia has grand ambitions in her latest book. When I first picked it up I confess I assumed that it wasn’t really aimed at me, a mathematician, and someone who is already reasonably well versed in formal logic. To some extent this is true of the first of three parts of the book, which surveys the essentials of propositional and quantifier logic. It does this however through the lenses of some of the hottest points of contention in modern social and political discourse: gone are the dreary predicates and facile conclusions of other treatise on logic, instead we learn implication by interrogating white privilege; negation by negotiating Brexit; and quantifiers by questioning sexism.

In a book which is ostensibly advocating for the broader use of abstract logic across all modes of discourse, part two comes as a refreshing riposte to that particular brand of mathematician who holds their subject as not merely superior to all others, but also claim its logical basis as an unimpeachable ziggurat. Cheng is not only forthcoming about the limits of logic, but actively embraces them as the way in to productively unifying the logical and emotional approaches to argument, discussion, and consensus.

This sets the stage for part three, where the promise of the subtitle is repaid. It provides a practical guide for how normal people (and indeed mathematicians) can actually apply ideas from formal logic to every day situations, as well as to interrogate their own beliefs and justifications for those beliefs. It is also an opportunity for Cheng to lay down her vision of the ideal format of interpersonal arguments and political discourse, and what it means to be an intelligently rational person. Contrary to the perhaps widely held belief that logic and emotion are antithetical – that the only things emotion can bring to an argument are fallacies – Cheng argues that logic and emotions not only can coexist in a rational debate, but must coexist. She writes “I think we can use this superpower [intelligent rationalism] to help the world bridge divides, foster a more nuanced and less divisive dialogue, and work towards a community that operates as one connected whole.”

Overall then I think the book succeeds in being approachable for someone with no training in formal logic, while still being engaging enough to sustain the interest of a professional mathematician throughout. From the start with an apparent oxymoronic title, at every stage Cheng manages to subvert the expectations of the reader as to the scope, applicability, and practicality of logic. The core message of the book, which if it can be reduced to a single word would be empathy, is hardly a novel one; the novelty comes from the framework with which Cheng approaches the topic, and this really makes this book stand out.

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A Compendium of Mathematical Methods

Jo Morgan is a maths teacher, poster of high quality teaching resources, tweeter, and organiser of excellent events. The second book on the Chalkdust Magazine Book of the Year 2019 shortlist is A Compendium of Mathematical Methods (Amazon UK, Waterstones, Foyles), a book written by Jo containing various methods for completing mathematical tasks that you need to do at school.

Style

This book presents multiple methods for performing common school-level mathematical tasks, such as subtraction, multiplication, simplifying surds, and polynomial division. These range from the most common methods that most readers will be familiar with to more unusual and unknown methods (often due to being older and not longer fashionable).

Control

The book presents multiple examples of each method alongside very clear explanations of the what is being done. For the less familiar methods, explanations of how the method works are given, alongside some historical or geographic context of where any why these methods were/are used.

Damage

This book presented many methods that I was unfamiliar with, and I had lots of fun working out how these methods related to methods I was more familiar with and how they worked. I particularly enjoyed the polynomial division section, where methods equivalent to polynomial long division, but with steps in which it was much more clear what was actually being done to the polynomial, were shown.

Aggression

I would strongly recommend this book to my maths teacher friends, and wish I’d had a copy of it when I used to be a maths teacher myself. I’m less likely to recommend it to my non-teacher friends, as the book has a clear teaching focus, but I know a few non-teachers who would find the discussions of different arithmetic methods interesting.