# Algorithms and Sloped Tax

This piece will start off about astronomy but segue into fair taxation.  You may have noticed that I have to keep writing about astronomy because I’m expected to, but I really want to write about all kinds of other things, so I have to find stratagems to do so.

Here is one of the charts I’ve been making toward a book; it shows the curious path of Neptune in the years 1862 and 1863.  (View it larger if you can.)

At that time Neptune, advancing along its vast 165-year orbit, crossed the celestial equator into the sky’s northern hemisphere; and, as seen from Earth, the equator-crossing was a quintuple event.  This was because the retrograde loop that Neptune appears to trace as we overtake it is long enough that each year’s loop overlaps the next; and this overlapping caused Neptune to cross the equator five times.

It was while making this particular chart that I decided to add to my charting program a way of marking the opposition of a planet: the moment in the year which is the climax of the planet’s time of observability.  This would save some of the labor of adding labels by hand (automating of labor is one of the urges for programming).  I should be able to generalize this later for other kinds of event, such as Venus’s greatest elongations.

Well, these opposition labels were more conveniently written on the south side of Neptune’s path, because that path is here convex toward the south.  (Though the curvature is slight, because Neptune isn’t far from the ecliptic.)  But what about other times at which the loops are convex toward the north, because a planet is traveling on the north side of the ecliptic plane?  In anticipation of this, I had to add a bit of programming to find whether this is so.  But how is the program to know whether the curve it is drawing from date to date is convex and in which direction?  This is an example of a situation in which you have to find an algorithm, that is, a pattern of thought that you can turn into a procedure.

Algorithm is an interesting word.  I tried to write my definition of it, just given, before seeing how the Oxford English Dictionary defines it, because I’ve so often had the experience of wondering “How on earth would I define sententious or tease or countryside or affectation or or or whatever or Well! or after all?” and finding that the dictionary’s experts hit the definitional nail on the head.  For algorithm the dictionary doesn’t do a whole lot better than I did.  It’s clear what algorithm means, but not easy to put into words.

It originated from the name of Abu Ja`far Muhammad ibn Musa al-Khwarizmi, a mathematician, astronomer, and geographer who worked in Baghdad but came from Khwârazm or Khwârizm.  That is one of the words spelt in Persian with a w which, after kh, is either sounded as v or dropped out, or sort of dropped out.  The region was in earlier Hellenistic times called Chorasmia, and is the oasis where the great river called Oxus in Greek, Amu Darya in Persian, comes out into the now nearly dried-up Aral Sea, in what is now Uzbekistan.  Khwarazm was the base of an Iranian civilization and of an empire covering much of central Asia and Iran, until destroyed by Genghis Khan the Mongol in 1220.

Al-Khwarizmi flourish four centuries before that, in the early 800s.  He more or less founded algebra, and the word derives from one of his algebraic operations, which he called al-jabr, “the restoring” (of a missing piece) and included in the title of one of his treatises.  He did not invent the Hindu-Arabic system of numerals, with zero and decimal places, but his work, translated from Arabic into Latin, introduced it to Europe, where it was called algorismus.  And the Spanish and Portuguese words for “digit” are guarismo and algarismo.  And do we suspect that arithmetic is yet another corruption of al-Khwarizmi’s name?  No, it’s from Greek, arithmos, “number.”  But compounded from that and logos, “word,” is logarithm, which by delightful coincidence is the anagram of algorithm!

From meaning the improved notation for arithmetic, algorism or algorithm moved, in the 1800s, toward a new meaning: a logical insight on which you can build a set of steps which should work exactly to achieve what you want – as in a programming language.  The algorithm underlies the steps of the program.  Recently, it has spilled over into being a buzz-word for procedures such as those devised by card-sharpers and financiers to beat the system.

So, how do we get a program to know which way a moving body’s path is curving, while it is drawing it as a series of small straight lines?  Here’s my algorithm.  If it knows the coordinates not only of the present point, but of the points a day earlier and a day later, then the line from the day before to the day after passes to the left of the present point if the curve is convex to the right.  So we make it calculate the angles from day-before to day-after and from day-before to present, and if (assuming angle increases counter-clockwise) subtracting the second angle from the first gives a positive number, the curve is convex to the right.

Of course.  Some algorithms are easy to see, after you’ve seen them.

Here’s another matter requiring an algorithm.  I read in a newspaper of October 12 that the election manifesto of Britain’s Labour party proposes “a 45% tax band on those earning more than £80,000 and a 50% rate for those on more than £123,000.”

Income tax systems are structured in steps, which are called “brackets” or “bands.”  If you earn more than X1, you pay Y1 percent; if more than X2, Y2 percent…  We could refer to the borderlines proposed in the manifesto as “thresholds” 80,000/45 and 123,000/50.  (I apologize for using such pluty incomes as examples, but they are the only ones I have at hand; I don’t at the moment know what actual thresholds there are in the American and British tax systems.)

It seems unfair that someone earning a penny over one of these thresholds pays substantially more than someone earning a penny under.  So people worry about the thresholds and cheat at them.  If you fear your income has been near one of them, you don’t know, till your accountant informs you and it’s too late; or, to make sure, you may resort to some “creative accounting.”

I long wondered: instead of this system of steps, couldn’t there be a slope?

In this graph of income vs. tax rate, you can think of “income” as dollars, pounds, or “ackers” (as British soldiers used to refer to whatever dinars or rupees were used in the countries they were sent to); it’s only for illustration.

Small circles represent the two sample thresholds I read of – 45 percent tax if your income is 80,000+, 50 percent if it’s 123,000+.  If your income is in between, your tax percentage stays on 45, as shown by the stepped line.

But the dots follow a slope pinned to the two thresholds.  In a sloped system, the rate remains proportional to actual income.

I thought it would be fairly easy to find the algorithm giving this slope, and my first idea was to get the angle of the slope and use it to calculate other points along it.  That became complicated; and you wouldn’t want to use trigonometry to figure your taxes.  The algorithm had to be based on those four given figures, which we can call income 1, rate 1, income 2, and rate 2 (80,000, 123,000, 45, and 50).  Rate has to increase in proportion to the fraction of the distance between the threshold below and that above.  That’s the algorithm (not very strictly stated), and the equation resulting from it is:

rate = rate 1 + (rate 2 – rate 1) * (income – income 1) / (income 2 – income 1)

Not quite simple, but only one equation, and simple compared with trigonometry.

It could easily be built into an app; or, rates could be read off from the dots on an enhanced version of the graph.

The formula could give a whole slope, below and above the thresholds, as indicated by the continuations of the dotted line.  But rules would be added that the rate switches to zero below a certain income, and stays flat above some top level, so as to avoid becoming 100 percent.

Moreover, there would probably be other thresholds lower down, representing decisions about what is fair taxation at various levels.  A fictitious example is added to the graph, in blue.  A tax system could consist of a series of slopes from threshold to threshold.

You will notice that, since dots in the graph are lower than the steps, almost everyone in the sloped system will be paying tax at a lower rate than in the stepped system – if the slope is kept where it is.  But the whole slope would be slid to the left so as to adjust for this.

Isn’t it fair that taxes should go up without jumps?  Maybe you know of some reason why it isn’t.

A couple of footnotes about the abstractified substance the love of which is “the root of all evil” (First Epistle to Timothy, 6:10):

An addition to my list of scummionaires has to be Paul Manafort, who financed a son-of-a-dictator spending style by running a P.R. company for right-wing despots (“the torturers’ lobby”), became chairman of the Trump election campaign, and has just now been revealed as having laundered the foreign sources of his money and hiding it from taxation.

Income inequality has risen to its highest since the corca-1905 “gilded age” or age of robber barons.  Globally, there are now 1,542 dollar billionaires; in 2016 they increased their wealth by almost 20 percent, to a record total \$6,000,000,000,000 (more than twice the gross domestic product of Britain).  Illionaires (under whom we could embrace those of the mil, bil, or tril kinds) are becoming somewhat fearful of a backlash of anger.  The “philanthropy” by which they try to make themselves liked consists mainly of building art galleries and buying sports teams.  The International Monetary Fund says governments had better force the top one percent to pay more tax.

## 17 thoughts on “Algorithms and Sloped Tax”

1. Tony Pablo says:

Hello Guy!!

This is Tony, we are the California couple you met at the restauarant in Chania (Tamam). I find your blog quite an interesting read. We are back now in Germany, and hopefully Mildenhall, UK before we return back to the states. Regards to you and wife Tilly.

Tony and Leoni Pablo

2. Curt says:

This algorithm is similar to one I devised when working on radar “filter regions” for air defense systems. An operator creates a region consisting of a sequence of ‘n’ points forming an area that is required to be geometrically convex. First, the region itself must be validated. Two points determine a line. So beginning with the first two points (P1 and P2), does the next segment (P2 and P3) bend to the “left” or “right” of the P1P2 line? This is determined using a “dot product” between P1P2 and P2P3. The dot product result is either positive, negative or zero (colinear points). The test continues with points P2, P3 & P4, and each segment through Pn-2, Pn-1 and Pn. Other than 0, all segments must have positive or all negative dot products. If not, then the figure is not geometrically convex and is rejected.

The test for whether or not an aircraft is in the filter is similar. If the aircraft is to the left or right of each segment of the filter then it is inside of the filter. Otherwise it is outside.

As far as taxes go, the objective is not for the government to attempt to collect as much tax ashumanly possible from the citizen. It is for the citizen to minimize the amount of their own property required to be absconded by the government.

3. Howard Wilk says:

The only thing I know about the UK tax system is that the top marginal income tax rate is–or used to be–95%. Reference: “Taxman”, the Beatles (written by George Harrison): Let me tell you how it will be, there’s one for you nineteen for me, cause I’m the taxman, yeah, I’m the taxman. Should five percent appear too small, be thankful I don’t take it all, cause I’m the taxman, yeah I’m the taxman…” Great song.

4. Joseph Koskie says:

Guy,

Taxes used to be a nightmare for me until I bought my first TurboTax, the greatest tool ever invented to fight governmental tyranny.

Joseph

1. Howard Wilk says:

PS I also recommend the contemporaneous “Sunny Afternoon” by the Kinks: “The taxman’s taken all my dough…”

5. John Flanigan says:

Aloha Guy

Your writings are always interesting. In “Algorithms and Sloped Tax” you did a nice job of presenting the source of the word. But you elided over the history of change of pronunciation from “algorism” (an “-ism”) to “algorithm.”

In my 1971 version of the OED after a long entry for “algorism” there is this single line:

“Algorithm, erron, refashioning of Algorism.”

When I first got interested in computer science c.1965, the books I read used the “-ism” form. As computer subjects became popular, I more and more started seeing the “erroneous” “-ithm” form used. I felt like I was being forced to lisp!

(I have a similar gripe with the (mis)pronounciation of “giga-” as in “gigabyte.” Engineers have used the soft “g” for words like “gigahertz” and “gigawatt” (jigawatt, jigahertz) since forever. Why did the mispronunciation by untutored newbies in the computer business cause us to change the pronunciation to the hard “g”? Doesn’t “gigantic” come from the same root? Pronouncing that with a hard “g” sounds really strange.)

I have no problem with English changing to accommodate necessary new ideas, but I resist supporting change brought about simply because of ignorance of linguistic history.

Thanks for giving me the opportunity to rant. Love your posts!

Me ke aloha pumehana,

1. A good enrichment to the etymological story.
It occurred to me that “algorism” was erroneously refashioned to “algorithm” on the analogy of “logarithm”, which was invented by Napier in 1614.
However, the OED online shows that Latin “algorithmus” and French “algorithme” date back to the 15th and 16th centuries.
I suspect that “algorism” was erroneously felt to contain the “-ism” morpheme, though that did not have any effect on etymological developments. (“Algorist”!)
John is my wonderful ally in the advocacy of Approval Voting.

6. Marcia L. Barr says:

Howard Shaw, that seems to be the North American solution: it’s used in Canada too. There is one rate starting at a low level of income and covering income up to a specified amount, then a higher rate for amounts between that and the next limit, and on and on, with the higher rates paid only on the amounts contained within the applicable brackets. The tax forms conveniently provide the amounts payable on filled brackets, so you have to do the multiplication only on the amount in the last partly empty bracket. In my pre-calculator youth, I did the arithmetic by hand, and just as I retired had progressed to having my husband do my taxes along with his using a tax calculation program. Some things have improved!

7. Jim Kelley says:

How does your algorithm for curvature handle labeling an opposition when a planet crosses the ecliptic very near opposition such as Mars in 1984? That opposition was so close to the ecliptic that there was a transit of Earth as seen from Mars. The points for the day before opposition, day of opposition and day after should be extremely close to a straight line.

1. A moving body’s path is unlikely to be exactly straight, when quantities are in double precision. In any case, my program avoids that by giving one of the others as default (roughly, “put the label below; if curvature is convex in the other direction, put the label above” – no mention of “if the line is straight!).

8. John Stroughair says:

If I can add a second comment it is somewhat misleading to compare the wealth of the billionaires with the GDP of the UK. GDP measures income not wealth. The total wealth of the UK is harder to measure than GDP but is likely around £20 trillion – so comfortably more than the figure for all the billionaires.

9. John Stroughair says:

There are no sudden jumps in the UK tax system. Somebody in your example earning a pound more than £80,000 pays the lower rate on £80,000 and then 45% of the extra pound.
So if the lower rate is 40% the person earning £80,000 pays £32,000 in tax; the person earning £80,001 pays £32,000.45 in tax.
You do get weird jumps when allowances and benefits get withdrawn at threshold levels, but that is a more complicated situation than just looking at the tax code and mostly affects the poorest members of society.

10. Mike says:

A slope would only be a set of smaller, or more, incremental steps. Our present system only has 3 or 4 steps, so the American public can “understand” the percentages. Then they trust an accountant to do their taxes for them, without ever bothering or trying to truly understand the system. Go TurboTax!

11. Howard Wilk says:

I would assume the tax system in the UK is similar to that in the US, in which you can’t take home less money on more income because you’ve moved into a higher tax bracket. Sayeth wikipedia: A marginal tax rate is the tax rate an individual would pay on one additional dollar of income. Thus, the marginal tax rate is the tax percentage on the last dollar earned. In the United States in 2016, for example, the highest marginal federal income tax rate was 39.6%, applying to earnings over \$415,050. Earnings under \$415,050 that year had a lower tax rate of 35% or less.

To right-wing and libertarian types who complain about high taxes I point out that in the 1950s, with a Republican President (Ike) and Republican control of both the House and the Senate, the highest marginal income tax rate was 91 or 92%.

12. Steve Meyer says:

Guy, I always enjoy both your astronomy facts and also your ruminations on other topics. Also,there’s long been a large sign painted on a building in my town: “Fight Unfair Taxes.” My observation is that taxes are neither fair or unfair — they just are.

13. Kevin B says:

One would suppose you are equally disposed to the Hillary & Bill Clinton Crime Syndicate’s style of international bribery…?
Certainly I would think that everyone of your readers feels the same way you do about Paul Manafort. Why then, would a comment on that be necessary – or even pertinent…?

14. Howard Shaw says:

I don’t know how they do taxes in Britain, but I’d be surprised if they didn’t use the “American” system. Something like 45% of the first 123000, then 50% of the amount OVER 123000 (numbers chosen for illustration only), which makes for the tax bill being “continuously” dependent (i.e., no sudden jumps) on income. Am I missing something?