Math Games

Geometric Lathes

Ed Pegg Jr., January 26, 2003

Chances are, you are very close to a Guilloche pattern.  If your computer ever had Windows on it, the Microsoft sticker on the side of the machine will have a Guilloche pattern.  If you have paper banknotes, you have a Guilloche pattern.  If you have a passport, or certain other identification cards, or a diploma, or a check or bond of any sort, you likely have a Guilloche pattern.

Figure 1.  Detail from a United States dollar bill.

We don't have pictures of the lathe and to my knowledge it is not on display.  I'll have to check to see where it is kept.  Please remember to forward the name of your magazine and provide the publication information.  Thank you.  Here is the response to your inquiry.   At one time all Bureau of Engraving and Printing (Bureau) guilloche (cycloid) patterns were done on a guilloche machine, or what the Bureau knew as a geometric lathe, which had many wheels (probably up to ten).  The "codes" to each pattern generated were known only to the Sculptural Engraver who operated the lathe.  For many years, that lathe was the source of all the lathe and cycloid work on U.S. paper currency.  Today, that work is all computer generated.
The Bureau does "compare notes" with other Banknote houses/companies around the globe.  There are images of most of the Bureau's earlier guilloche (cycloid and lathe) patterns in our files; however, these images are security items and are unavailable, primarily because they are still "usable" in our currency design.  A great amount of the design "parts" of the redesigned Series 1996 currency (the series of currency where the portraits were enlarged and moved off center) came from the lathe images in the Bureau files.
I hope this adequately responds to your inquiry.  Also, please let me know the official name of your magazine publication and when or if this information will be published.

Since October 1, 1877, all U.S. currency has been printed by the Bureau of Engraving and Printing, which started out as a six person operation using steam powered presses in the basement of the Department of Treasury. Now, 2,300 Bureau employees occupy twenty-five acres of floor space in two Washington, D.C. buildings.

Of all the world, the site with the best pictures of its own currency seems to be the Bank of Russia.  Here is a 1000 Ruble banknote.

Albanian Passport:
American Currency Exhibit:
Amgraf Security Documents:
Artlandia Symmetry Works Guilloche patterns:
Central Bank of the Russian Federation:
Excentro: Guilloche design generator for Macintosh:
National Bank of Austria:
National Bank of Denmark:
Norges Bank:
Reserve Bank of New Zealand:
Ron Wise World Paper Money:
Security Graphics:
US Bureau of Printing and Engraving:
Vivian Alexander's Guilloché technique:

In 1936, Harold Cramer conjectured that 1 > gap(p)/ ln(p)^2 for all primegaps. Figure 1 shows how Cramer's conjecture is doing as of 2004, with a plot of gap(p)/ln(p)^2 for the first 67 maximal primegaps.  As you can see, the graph is approaching 1. 1132/ln(1693182318746371)^2 ~ .920639 is the high point.

Cramer Conjecture
Figure1.  A plot of Cramer's Conjecture.

A list of currently unverifiable probable primes is maintained at

So, arbitrarily large gaps appear between prime numbers.  On the complex plane, does the same apply to Gaussian primes?

Gaussian Primes
Figure 2.  The Gaussian Primes

Yes, there are arbitrarily large prime-free regions on the complex plane. Define a+bI# as the product of all Gaussian primes n+mI satisfying 0≤an and 0≤bm. Define a+bI! as the product of all Gaussian integers n+mI satisfying 0≤an and 0≤bm.  A prime-free block of Gaussian integers can be found. Just like with regular primes, prime-free blocks with smaller number ranges can be found.

I close with three problems.

Puzzle 1: Divide the numbers {2 3 5 7 11 13 17} into two sets that have products n and n+1.
Puzzle 2: Factor two consectutive numbers higher than {5909760,5909761} into numbers all smaller than 20.
Puzzle 3: What is the smallest 6×6 prime-free Gaussian integer block?


Cramer, Harold "On the order of magnitude of the difference between consecutive prime numbers," Acta. Arithmetic, 2, 23-46. 1936.

Lifchitz, Henri. Probable Prime Top 5000.

Martin, Marcell. Primo Primality Proving.

Nicely, Thomas R. First Occurrence Prime Gaps.

Peterson, Ivars. MathLand: Prime Gaps.

Wagon, Stan. Mathematica in Action, 2nd edition. Springer-Verlag, 1999.

Weisstein, Eric W. "EllipticCurvePrimalityProving.", "PrimeNumberTheorem." From MathWorld--A Wolfram Web Resource.,

Mathematica Code for Figure 1:

primegaps = {{1,2}, {2,3}, {4,7}, {6,23}, {8,89}, {14,113}, {18,523}, {20,887}, {22,1129}, {34,1327}, {36,9551}, {44,15683}, {52,19609}, {72,31397}, {86,155921}, {96,360653}, {112,370261}, {114,492113}, {118,1349533}, {132,1357201}, {148,2010733}, {154,4652353}, {180,17051707}, {210,20831323}, {220,47326693}, {222,122164747}, {234,189695659}, {248,191912783}, {250,387096133}, {282,436273009}, {288,1294268491}, {292,1453168141}, {320,2300942549}, {336,3842610773}, {354,4302407359}, {382,10726904659}, {384,20678048297}, {394,22367084959}, {456,25056082087}, {464,42652618343}, {466,127976334671}, {474,182226896239}, {486,241160624143}, {490,297501075799}, {500,303371455241}, {514,304599508537}, {516,416608695821}, {532,461690510011}, {534,614487453523}, {540,738832927927}, {582,1346294310749}, {588,1408695493609}, {602,1968188556461}, {652,2614941710599}, {674,7177162611713}, {716,13829048559701}, {766,19581334192423}, {778,42842283925351}, {804,90874329411493}, {806,171231342420521}, {906,218209405436543}, {916,1189459969825483}, {924,1686994940955803}, {1132,1693182318746371}, {1184,43841547845541059}, {1198,55350776431903243}, {1220,80873624627234849}}

ListPlot[Map[First[#]/Log[Last[#]]^2 &, pr], PlotJoined -> True, PlotRange -> {0, 1}];

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Ed Pegg Jr. is the webmaster for He works at Wolfram Research, Inc. as the administrator of the Mathematica Information Center.