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The math behind Apollo 11


Buzz Aldrin on the lunar surface, July 20, 1969, photographed by Neil Armstrong

It’s been over three years since I last posted to this blog. Not that I stopped writing. In addition to my long running monthly Devlin’s Angle for the Mathematical Association of America, early this year I launched the Stanford Mathematics Outreach Project (, which has its own blog. The former is aimed, fairly broadly, at high school on college level mathematics instructors and, to some extent, their students. The latter focuses on K-12 mathematics teachers, and was designed to complement, and collaborate with, Stanford University Mathematics Education Professor Jo Boaler’s youcubed project.

Having retired from full-time employment at Stanford at the end of last year, however, I find I have time to reflect on other mathematics related issues that don’t entirely fit on either of those blogs. Time then, to revive this outlet. To be sure, it was frequently a toss-up whether to post on or Devlin’s Angle, and the same has already occurred with the SUMOP blog. What tipped the scales in favor of reviving this blog was the upcoming Fiftieth Anniversary of the Apollo 11 moon landing on July 20, 1969.

On June 23, CNN broadcast a new documentary of the Apollo mission, using video footage never previously seen. You can watch the whole thing (just under two hours), for a small charge, on YouTube or Amazon Prime. For reasons that will momentarily become clear, I had to watch it at the first opportunity. So I joined the many others (mostly around my age, I suspect) to view the first screening. Soon after the movie started, I decided to live-tweet my reactions. The remainder of this post is a transcription of that twitter stream, with the inevitable twitter-typos cleaned up. [For the record, the original twitter stream can be found here.]


Watching the CNN documentary about Apollo 11. My context. At the time, in July 1969 I was a 22 year-old, English speaking, white male with a new math degree. Apollo was an incredible achievement for the group I was a de facto member of 1/

I was fluent in all of NASA’s languages: English, mathematics, white male machismo. By 2001, when I wrote my book The Math Gene, my understanding of math had increased to appreciate its role in the creation and management of human teams  2/

Here is the PROLOGUE of The Math Gene 3/

In fact, as I have said often, I became a mathematician because I wanted to be part of the space era. See for example my recent video interview 4/

Had I been born just a few years earlier, to American parents rather than English, I would have done all I could to be one of those math and engineering geeks at Johnson Space Flight Center whose names were at the time as familiar to me as were those of my family 5/

Watching that video footage today, now as a holder of a US Passport and with several years of service as a mathematician to three organizations in the US Department of Defense, my reactions are complex. 6/

The thrill is still there. I would have loved to have been part of it, not merely an avid follower from the other side of the Atlantic. After I came to Stanford in 1987, with NASA Ames Research Center just down the road, I met a few of those who were involved. 7/

Moreover, some of my research found applications at NASA in the later missions of the International Space Station. But, though they involved way more complex engineering, they lacked the sheer drama of Apollo 11, … /8

… a mission that was conducted entirely by following a carefully worked out mathematical plan, in an environment where any one of millions of possible small miscalculations would bring the mission to a tragic end 9/

But here in 2019, I watched the video with other emotions. One was the degree to which Apollo 11 was, at least in its public presentation, almost exclusively a white males operation. I saw just one woman and one black male in one of the control rooms. 10/

Of course, as everyone knows thanks in large part to the movie Hidden Figures, NASA’s reality was far from that, with African American women mathematicians in particular playing a crucial role from the very early missions on 11/

That American society has moved on since the 1960s (at least in part) was brought home vividly by the dramatic contrast between the all-white-males video footage and the commercial breaks, with their carefully selected representatives of modern US society 12/

But America has also moved backwards. Instead of taking inspiration from those in the Apollo era, and seeking to do more great things, but this time in a far more inclusive way, we have an entire major party whose goal is to MAGA by going back to the 60s social divisions 13/

To do this, they try to kill anything that depends on more than 1960s scientific knowledge, and insist we do things the way they were back then.  But that is not what Apollo was about!!! 14/

Apollo was about taking a bold step into the unknown. To join together and move forward into the future – “to boldly go …”, as the best known fiction of space exploration famously proclaims 15/

It doesn’t have to be space. It could be anything we choose. To those of us who felt “in the club” (i.e., white , male, math-geeky, and fluent in English), Apollo was a thrilling life experience. How about creating another one, but this time one that includes all Americans? 16/

In education? Health care? Curing famine? Addressing climate change? There’s no shortage of big goals. Indeed, those four tower above a Moon landing in terms of importance to society. All it takes is the will. Apollo happened only because President Kennedy expressed the will 17/END

Steve Jobs remembered

The fact that I, along with millions of other people, learned of Steve Jobs’ death by way of a text message or email sent to my iPhone indicates just how huge was the Apple co-founder and CEO’s impact on the way many of us go about our business and live our lives. I can’t imagine any other CEO of a large corporation whose death would be so widely mourned.

By chance, the Apple Store in Job’s home town of Palo Alto, where I live, is right across the street (University Avenue) from the Peet’s Coffee Shop where I go for my morning latte. As I left the coffee house the morning after his death was announced, I noticed some flowers and a couple of lighted candled had been left on the sidewalk in front of the Apple Store window. I walked over and took a couple of photos. A day later, that small shrine had grown considerably.

Since Jobs had lived less than a mile from my home, on the Saturday morning I detoured past his (surprisingly modest) suburban-style house on my way to the supermarket, where I found precisely what I had anticipated. A huge outpouring of grief and remembrances.

Again it struck me: this reaction to the passing of a billionaire CEO? I cannot imagine a similar response to the death of Bill Gates, Larry Page, Sergey Brin, Mark Zuckerberg, or any of the other technology elite. Somehow,  Jobs did more than provide people with useful consumer products, he touched their emotions.

To be sure, not everyone is an Apple fan. But to those who are, their attachment to the brand is deep. People tend not to like Apple stuff, rather they LOVE it. And that is surely a tribute to superb design – design both for appearance and for usability.

I can’t say I’ve ever been a fan of anyone or anything, but I am a sucker for great design. When I first saw, and used, a Macintosh (one of the first generation), it quite literally changed my career, and in due course my life. As a mathematician with research interests in computation and a side interest (more of a hobby) in computational number theory, I was using computers long before the personal computer came on the scene. Back then (and this was the 1970s), a computer was a slave that you commanded to do things for you. Even worse, it was a dumb slave, so you had to formulate those commands with extreme care. A single missing or misplaced comma would cause the machine to grind to a halt.

All that changed with the Mac. As a user, you were no longer issuing commands to be acted upon, you were HANDLING INFORMATION. It was your stuff, and you were the one performing the action. Or so it seemed. In reality, you were being fooled by a cleverly designed interface.  But that was the genius of the Mac; it took something intrinsically alien to human beings, computation, and presented it is a way that we find natural and instinctive.

Well, actually, that genius was not Apple’s, rather that of a remarkable group of researchers at the Xerox Palo Alto Research Center (PARC) who had developed this new approach to computing over several years. To Xerox (and many of the researchers at PARC), the goal was to build computers to support office workers. Jobs’ genius was recognizing that, properly packaged and marketed, this approach to computing could turn computers into mass-market consumer products.

When I first started to use a Mac, I realized that there was huge value to be had in viewing reasoning not as a PROCESS of logical deduction to arrive at a conclusion (the classical view of mathematical reasoning), but as  GATHERING INFORMATION in order to reach a decision. The result of that shift in viewpoint first resulted in my book Logic and Information, which was published in 1990, and my entire research career since then has followed on the heels of that book.

Were it not for the Mac, I doubt I would have shifted my research the way I did. It would not have been enough to read about the WIMPS interface (windows, icons, mouse, pointers), or even to watch someone else using it. It was the powerful sensation of DOING it yourself, of physically manipulating items of information, that made all the difference. The Mac was not a device you used. It was something you experienced. And that is a profound difference.

As became clear through his entire career, Jobs had a deep appreciation for the importance of making technology something people experience – not merely use.

By pure chance, just two months before Jobs died, I published a short e-book in which I compared him with one of the greatest innovators of all time, the thirteenth century mathematician Leonardo of Pisa, known more commonly today by the modern nickname Fibonacci. Leonardo changed the world by writing a book that introduced modern arithmetic to the western world.

The stories of what these two men did exhibit remarkable similarities. Leonardo packaged arithmetic and, through the medium of parchment, gave personal computing to the masses. Jobs made personal computing accessible to everyone through the medium of silicon. Neither individual was an inventor. Their genius was taking something alien and complex and making it accessible – and friendly – to all.

For more details on the Jobs – Leonardo connection, see my MAA column “Devlin’s Angle” for August of this year.

Apple homepage October 6, 2011


I’m Dr. Keith Devlin, an emeritus mathematician at Stanford University, an author, the Math Guy on NPR’s Weekend Edition, and an avid cyclist. (The header photo is me halfway up Mt. Baldy in Southern California.)

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