The final completion of my 2013 calendar, Graphical Astronomy, has been delayed, so at this point I am going to update the dates for 2014 and post it this fall. As partial compensation, I’ve created a Valentine’s Day card for mathematically-inclined people that can be downloaded, printed and folded. It is appropriate whether the person giving the card or receiving it is interested in math, or both, and in fact it’s not Valentine’s Day specific so it can be used for birthdays or anytime at all.
Using a cardioid to represent a heart is not new, but I re-created the complex construction on the front from an old doctoral dissertation of 1900 by Raymond Clare Archibald at Kaiser-Wilhelms -Universität Strassburg. I plotted the curves in vector form in Mathematica and added the lines and text in Adobe Illustrator. I stylized the cardioid in the center of the figure by plotting several versions of it where I added a small parabolic term to the equation so for each curve the width of the cardioid in the middle varied. The straight lines are gray, actually, so the figure will be less cluttered when printed than when you view it normally in the PDF (you can zoom into the PDF to see this).
The image below provides a low-resolution view of the outside and inside of the card before folding down the middle, although the gray background is not included in the actual PDF. As you can see from the inset picture at the top of this essay, the curves and lines on the front side are much sharper then they appear here. The link downloads the vector-resolution PDF with rotated images that you would print in double-sided Portrait mode to line up the outer and inner images correctly for a side-opening card. Of course, it looks best on photo paper or glossy color cardstock, and perhaps at a bit smaller size than 8.5″x11″. It should scale appropriately for A4 or other paper sizes. You will probably want to write the event on the front and write something on the inside.
There is no indication on the card of its origin. If you ever want to know where you downloaded it from, you can find my contact information in the PDF file under the File—>Properties menu option of Adobe Acrobat. And if you would like something different printed inside or on the front, just write me using the Contact tab along the top and I’ll be happy to customize it for you.
— Ron
Download PDF of Valentine’s Day Card
P.S. I stumbled across a nice image for a music-themed Valentine’s Day card: The sheet music of Baude Cordier (c. 1380-1440) titled Belle, Bonne, Sage (Beautiful, Good, Wise), seen on the right and downloadable from here. According to Wikipedia, the red notes in this love song indicate rhythmic variations.
Dead Reckonings 
Glad you’re back! looking forward to the calendar
Thanks, Eric. I’ve never really left, but I’ve been so busy on projects this past year, including co-authoring a book on nomography, that I’ve had little time to post much. I’m really going to make an effort to finish up some semi-finished posts and be more active here. I worked hard to finish the new calendar on time, but it will be another week or so and then it’s already the end of January. I may expand some of the calendar page topics into individual essays here, though. — Ron
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Very nicely done – crisp, polished and clever
Thanks! — Ron
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Yes, welcome back, long time no hear!
Very nice card, which will be downloaded and sent to my wife for her birthday, which is also Valentine’s day. 🙂
Gordon
Thanks, Gordon! That’s a very convenient coincidence. — Ron
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Happy to see any post :o)
>> “…including co-authoring a book on nomography, …”
Title?
When will it be available?
Hi Scott! Your comment got me wondering, and I just went back and found that I apparently never completely responded to your email about Grimes’ article on medical nomograms and your comments comparing nomograms with other calculating methods. I will contact you again by email this weekend. Sometimes when corresponding with thoughtful people I start gathering information to write a thoughtful response and I get so busy that I forget to follow up.
I’ve also been intending to write an essay on the book, too, which came out last spring. I’m going to try to post my review on this blog in the next week. Back in 1912 the mathematician Thomas Hakon Gronwall published a paper that for the first time provided the necessary and sufficient condition for a three-variable equation to be represented by a nomogram. Unfortunately, the condition was that a solution existed to a pair of complicated partial differential equations. A general solution to this equation has been an effort in the field of web theory ever since, corresponding to linearizing 3-webs, and may have been finally solved within the past decade or so. But Gronwall took particular cases of this equation that he was able to solve explicitly, and in the process categorized types of nomograms (including conical nomograms first introduced by J. Clark in 1904) and to introduce new types. Remember how you can manipulate a determinant equation to create different affine and projective transformations of a nomogram? These are called homographic transformations, and as one example Gronwall was able to show that the only type of 3-variable equation that had nomographic forms that can not be obtained from each other by homographic transformations is the simple sum f(u) + f(v) + f(w) = 0, and he was able to categorize the unique forms of nomogram for it.
Anyway, Scott Guthery of Docent Press (who published Evesham’s book on the history and development of nomography that I reviewed before) led the collaborative effort to write a book on Gronwall’s nomographic work and this paper in particular. Alan Gluchoff of Villanova University wrote a fascinating chapter on Gronwall’s life and how his work fit into the history of nomography, Paul Hamburg translated Gronwall’s paper into English from the French, and in addition to being the overall editor Scott compiled a comprehensive bibliography of Gronwall’s prolific work including his unpublished manuscripts. Gronwall’s paper is a dense parade of equations without any figures, so I wrote a commentary that describes what he was doing in more detail and uses his results to derive and create examples of the different forms of nomogram he categorizes, including the homographically-unique forms for the sum of three functions. I used PyNomo to create the nomograms, and since one of his derived forms for the simple sum is a Weierstrass elliptic curve, my zoomorphic fish nomogram finally saw print in this book! It’s not a book for learning how to make nomograms, as the mathematics will surely drive away newcomers, but a book about an interesting historical paper that illuminates the various classes of nomograms. In your case, I know you are quite knowledgeable about nomography and wouldn’t have any problem with it. Anyone who likes Evesham’s book on the history of nomography, which includes a section describing this very paper, would find this an interesting book in the history of mathematics, I think.
If you’re interested after all this, the book is called Calculating Curves: The Mathematics, History and Aesthetic Appeal of T.H. Gronwall’s Nomographic Work and can be found at http://www.amazon.com/Calculating-Curves-Mathematics-Aesthetic-Nomographic/dp/0983700435 . Thanks for asking, Scott! — Ron
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Thank you for such a wonderful, beautiful and delightful card. My wife endures my fascination with math and so she will see that this comes right from the heart.
You’re welcome, Keith. It sounds like your wife is a lot like my wife. The best thing about making this is that I won’t have to pore through card racks at the last minute this year. 🙂 — Ron
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Hi Ron,
Your card was a big hit. She loved it. We have it prominently posted on the mantle over the fireplace. Let me know what you have in store for Christmas.
Thanks again -Keith
Thanks for the feedback, Keith! I found that I had a package of 8.5″x11″ photo-grade cardstock pre-creased to use as greeting cards, with envelopes, and I was really happy with how it printed out. Christmas, that’s a novel thought. A fractal snowflake would be possible, but fractals have never interested me. Maybe a snowflake or star design by hypocycloids (like spirographs) or an ornament design made from astroids. I’ll definitely do something, and I’ll email you when I post it here next fall. Thanks again for taking the time to post such a nice comment. — Ron
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