I’ve seen the illustration of nesting golden rectangles many times, but I’ve never seen a presentation go into much detail. This post will go into more detail than usual, including Python code.

Start with a golden rectangle in landscape mode. We’ll plot our rectangle with the lower left corner at the origin and the upper right corner at (φ, 1) where φ is the golden ratio. No imaging cutting off the largest square you can on the left side, shown in blue.

Next, focus on the remaining rectangle and imagine cutting off the square on top, shown in green.

Now focus on the remaining rectangle and imagine cutting off the square on the right, shown in orange.

Finally, focus on the remaining rectangle and imagine cutting off the square on bottom, shown in red.

So we’ve cut off left, top, right, and bottom. We’re left with a rectangle with one side of each color above. We could repeat the process above on this rectangle, cutting off left, top, right, and bottom, and so on ad infinitum.

Where does this process end in the limit? Draw two diagonal lines as shown below.

Note that these diagonals cut the innermost rectangle in the same way as the outermost rectangle. So every time we repeat the four steps above, we will always have two diagonals that intersect in the same place. So the limit of the process converges on the point that is the intersection of the two diagonal lines.

The two diagonals lines have equations

y = -(1/φ)x + 1

and

y = φx – φ

and so we can calculate their intersection to be at

x₀ = (2φ + 1)/(φ + 2)

and

y₀ = 1/(φ + 2).

So the construction process, in the limit, converges to the point (x₀, y₀) where the diagonals cross.

Here’s Python code to product the final plot above.

    import numpy as np
    import matplotlib.pyplot as plt
    
    phi = (1 + 5**0.5)/2
    
    def plotrect(lowerleft, upperright, color):
        x0, y0 = lowerleft
        x1, y1 = upperright
        plt.plot([x0, x1], [y0, y0], c=color)
        plt.plot([x0, x1], [y1, y1], c=color)
        plt.plot([x0, x0], [y0, y1], c=color)
        plt.plot([x1, x1], [y0, y1], c=color)
    
    plt.gca().set_aspect("equal")   
 
    plotrect((0, 0), (phi, 1), "black")
    plotrect((0, 0), (1, 1), "blue")
    plotrect((1, 2-phi), (phi, 1), "green")
    plotrect((2*phi - 2,0), (phi, 2-phi), "orange")
    plotrect((1,0), (2*phi-2,2*phi-3), "red")
    plt.plot([0, phi], [1, 0], "--", color="gray")
    plt.plot([1, phi], [0, 1], "--", color="gray")
    plt.savefig("golden_rect5.png")

I intend to find the equation of the spiral that touches two corners of each square in the next post.

In my previous post I mentioned John Conway’s Doomsday rule for calculating the day of the week for any date. This method starts off very simple, but gets more complicated when you actually use it.

This post will present an alternative method that’s easier to use in practice and can be described more succinctly.

Here’s the algorithm.

1. Take the last two digits of the year and add the number of times 4 divides that number.
2. Add a constant corresponding to the month.
3. Add the day of the month.
4. Subtract 1 for January or February of a leap year for .
5. Take the remainder by 7.

The result is a number that tell you the day of the week.

To be even more succinct, let y be the number formed by the last two digits of the date. Let d be the day of the month. Let L equal 1 if the year is a leap year and the date is in January or February and 0 otherwise. Then the algorithm above can be written as

(y + ⌊y/4⌋ + d – L + month constant) % 7.

Here ⌊x⌋ is the floor of x, the greatest integer no greater than x, and x % 7 is the remainder when x is divided by 7.

I’ve deliberately left out a couple details above. What are these month constants, and how does the number at the end give you a day of the week?

Customizing for the 21st century

I learned the method above in the 20th century, and the rule was optimized for the 20th century. You had to subtract 1 for dates in the 21st century.

Also, when I learned the method, it numbered the days of the week with Sunday = 1, Monday = 2, etc. Now I would find it easier to start with Sunday = 0.

Here’s an updated table of month constants that eliminates the need to adjust for dates in the 20th century, and numbers the days of the week from 0.

    | January | 6 | February | 2 | March     | 2 |
    | April   | 5 | May      | 0 | June      | 3 |
    | July    | 5 | August   | 1 | September | 4 |
    | October | 6 | November | 2 | December  | 4 |

If you’d like to number Sunday as 1 rather than 0, add 1 to all the numbers above.

The article I learned this method from came with suggested mnemonics for remembering the month constants. But when you change the constants like I’ve done here, you have to come up with new mnemonics.

Examples

I’m writing this post on Saturday, May 7, 2022. Let’s verify that the method gives the correct day of the week today.

Start with 22, and add 5 because ⌊22/4⌋ = 5. The number for May is 0, so there’s nothing to add for the month. We add 7 because today’s the 7th. This gives us

22 + 5 + 7 = 34

which gives a remainder of 6 mod 7, so this is day 6. Since we started with 0 for Sunday, 6 is Saturday.

Next, let’s do January 15, 2036. We get

36 + 9 + 6 + 15 – 1 = 65

which is 2 mod 7. so January 15 will fall on a Tuesday in 2036. Note that we subtracted 1 because 2036 will be a leap year and our date is in January.

If we want to look at Christmas 2036, we have

36 + 9 + 4 + 25 = 74

which is 4 mod 7, so December 25 will fall on a Thursday in 2036. We didn’t subtract 1 because although 2036 is a leap year, we’re looking at a date after February.

Other centuries

For dates in the 20th century, add 1.

For dates in the 22nd century, subtract 1.

If by some reason you’re reading this article in the 22nd century, make your own table of month constants by subtracting 1 from the numbers in the table above.

More details

When I was at University of Texas, humanities majors had to take a class we called “math for poets.” The class was a hodgepodge of various topics, and the most popular topic when I taught the class was this method. Here’s a handout I gave the class.

That handout is a little slower paced than this blog post, and goes into some explanation for why the method works. (The method was meant to motivate modular arithmetic, one of the topics on the syllabus, so explaining why the method works was secretly the whole point of teaching it.)

The handout is still valid. You can follow it verbatim, or you can replace the table of month constants with the one above and ignore the 21st century correction step.

Related posts

• Calendars and continued fractions
• Memorizing numbers
• Mentally calculating common functions

John Horton Conway (1937–2020) came up with an algorithm in 1973 for mentally calculating what day of the week a date falls on. His method, which he called the “Doomsday rule” starts from the observation that every year, the dates 4/4. 6/6, 8/8, 10/10, 12/12, 5/9, 9/5, 7/11, and 11/7 fall on the same day of the week [1], what Conway called the “doomsday” of that year. That’s Monday this year.

Once you know the doomsday for a year you can bootstrap your way to finding the day of the week for any date that year. Finding the doomsday is a little complicated.

Conway had his computer set up so that it would quiz him with random dates every time he logged in.

Mental exercises

Recently I’ve been thinking about mental exercise rituals, similar to Conway having his computer quiz him on dates. Some people play video games or solve Rubik’s cubes or recite poetry.

Curiously, what some people do for a mental warm-up others do for a mental cool-down, such as mentally reviewing something they’ve memorized as a way to fall asleep.

What are some mental exercise rituals that you’ve done or heard of other people doing? Please leave examples in the comments.

More on Conway

The drawing at the top of the page is a sketch of Conway by Simon Frazer. The strange thing coming out of Conway’s head in the sketch is Alexander’s horned sphere, a famous example from topology. Despite appearances, the boundary of Alexander’s horned sphere is topologically equivalent to a sphere.

Conway was a free-range mathematician, working in a wide variety of areas, ranging from the classification of finite simple groups to his popular Game of Life. Of the 26 sporadic groups, three are named after Conway.

Here are some more posts where I’ve written about John Conway or cited something he wrote.

• Never applied for a job
• Opposite of Parkinson’s law
• Ternary Golay code
• Almost prime numbers and almost integers
• Subprime fib sequence

[1] Because this list of dates is symmetric in month and day, the rule works on both sides of the Atlantic.