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Changing one character in a PDF

Posted on by John

I saw a post on X saying

Changing a hyphen to an en-dash increases your PDF file size by ~10 bytes.

My first thought was that it had something to do with hyphen being an ASCII character and an en-dash not. Changing a hyphen to an en-dash would make a UTF-8 encoded text file a couple bytes longer. (See why here.) Maybe adding one non-ASCII character could cause the file to include a glyph it didn’t before.

I did a couple experiments. I made a minimal LaTeX file with only the text

    See pages 9-10.

and another with

    See pages 9--10.

(In LaTeX, a hyphen compiles to a hyphen and two hyphens compile to an en-dash.)

I compiled both files using pdflatex. The PDF with the hyphen was 13172 bytes and the one with the en-dash was 13099 bytes. Replacing the hyphen with the en-dash made the file 73 bytes smaller.

I repeated the experiment with a Libre Office ODT document. Changing the hyphen to an en-dash reduced the file size from 13548 bytes to 13514 bytes.

Then I went back to LaTeX and pasted a paragraph of lorem ipsum text into both files. Now the file with the hyphen produced a 18131 byte PDF and the file with the en-dash produced a 18203 byte PDF. So in this instance changing the hyphen to an en-dash increased the size of the file by 72 bytes.

After adding the lorem ipsum text to the ODT files, the PDF resulting from the file with the hyphen was 23 bytes larger, 17846 bytes versus 17823 bytes.

PDFs are complicated. Changing one character can make the file bigger or smaller, by an unpredictable amount. It depends, among other things, on what software was used to create the PDF. Even changing one letter in an all-ASCII text can change the size of the PDF, I suppose due to some internal text compression and aesthetic control characters. I don’t pretend to understand what’s going on inside a PDF.

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nth derivative of a quotient

Posted on by John

There’s a nice formula for the nth derivative of a product. It looks a lot like the binomial theorem.

(gh)^{(n)} = \sum_{k=0}^n \binom{n}{k} g^{(k)} h^{(n-k)}

There is also a formula for the nth derivative of a quotient, but it’s more complicated and less known.

We start by writing the quotient rule in an unusual way.

\left(\frac{g}{h}\right)^{(1)} = \frac{1}{h^2} \left| \begin{array}{cc} h & g \\ h^\prime & g^\prime \\ \end{array} \right|

Applying the quotient rule twice gives the following.

\left(\frac{g}{h}\right)^{(2)} = \frac{1}{h^3} \left| \begin{array}{ccc} h & 0 & g \\ h^\prime & h & g^\prime \\ h^{\prime\prime} & 2h^\prime & g^{\prime\prime} \\ \end{array} \right|

And here’s the general rule in all its glory.

\left(\frac{g}{h}\right)^{(n)} = \frac{1}{h^{\,n+1}} \left| \begin{array}{cccccc} h & 0 & 0 & \cdots & 0 & g \\[3pt] h^\prime & h & 0 & \cdots & 0 & g^\prime \\[3pt] h^{\prime\prime} & 2h^\prime & h & \cdots & 0 & g^{\prime\prime} \\[3pt] \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\[3pt] h^{(n)} & \binom{n}{1}h^{(n-1)} & \binom{n}{2}h^{(n-2)} & \cdots & \binom{n}{1}h^\prime & g^{(n)} \end{array} \right|

 

Source: V. F. Ivanoff. The nth Derivative of a Fractional Function. The American Mathematical Monthly, Vol. 55, No. 8 (Oct., 1948), p. 491

The Great Pyramid of Giza and the Speed of Light

Posted on by John

Saw a post on X saying that the latitude of the Pyramid of Giza is the same as the speed of light.

I looked into this, expecting it to be approximately true. It’s exactly true in the sense that the speed of light in vacuum is 299,792,458 m/s and the line of latitude 29.9792458° N passes through the pyramid. The exact center of the pyramid is at 29.97917° N, 31.13417° E.

Of course this is a coincidence. Even if you believe that somehow the ancient Egyptians knew the speed of light, the meter was defined four millennia after the pyramid was built.

Roman moon, Greek moon

Posted on by John

I used the term perilune in yesterday’s post about the flight path of Artemis II. When Artemis is closest to the moon it will be furthest from earth because its closest approach to the moon, its perilune, is on the side of the moon opposite earth.

Perilune is sometimes called periselene. The two terms come from two goddesses associated with the moon, the Roman Luna and the Greek Selene. Since the peri- prefix is Greek, perhaps periselene would be preferable. But we’re far more familiar with words associated with the moon being based on Luna than Selene.

The neutral terms for closest and furthest points in an orbit are periapsis and apoapsis. but there are more colorful terms that are specific to orbiting particular celestial objects. The terms perigee and apogee for orbiting earth (from the Greek Gaia) are most familiar, and the terms perihelion and aphelion (not apohelion) for orbiting the sun (from the Greek Helios) are the next most familiar.

The terms perijove and apojove are unfamiliar, but you can imagine what they mean. Others like periareion and apoareion, especially the latter, are truly arcane.

Artemis II, Apollo 8, and Apollo 13

Posted on by John

The Artemis II mission launched yesterday. Much like the Apollo 8 mission in 1968, the goal is to go around the moon in preparation for a future mission that will land on the moon. And like Apollo 13, the mission will swing around the moon rather than entering lunar orbit. Artemis II will deliberately follow the trajectory around the moon that Apollo 13 took as a fallback.

Apollo 8 spent 2 hours and 44 minutes in low earth orbit (LEO) before performing trans-lunar injection (TLI) and heading toward the moon. Artemis II made one low earth orbit before moving to high earth orbit (HEO) where it will stay for around 24 hours before TLI. The Apollo 8 LEO was essentially circular at an altitude of around 100 nautical miles. The Artemis II HEO is highly eccentric with an apogee of around 40,000 nautical miles.

Apollo 8 spent roughly three days traveling to the moon, measured as the time between TLI and lunar insertion orbit. Artemis II will not orbit the moon but instead swing past the moon on a “lunar free-return trajectory” like Apollo 13. The time between Artemis’ TLI and perilune (the closest approach to the moon, on the far side) is expected to be about four days. For Apollo 13, this period was three days.

The furthest any human has been from earth was the Apollo 13 perilune at about 60 nautical miles above the far side of the moon. Artemis is expected to break this record with a perilune of between 3,500 and 5,200 nautical miles.

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Morse code tree

Posted on by John

Peter Vogel posted the following image on X yesterday.

The receive side of the coin is a decision tree for decoding Morse code. The shape is what makes this one interesting.

Decision trees are typically not very compact. Each branch is usually on its own horizontal level, with diagonal lines going down from each node to its children. But by making the lines either horizontal or vertical, the tree fits nicely into a circle.

I thought for a second that the designer had made the choices of horizontal or vertical segments in order to make the tree compact, but that’s not so. The direction of the path through the tree changes when and only when the Morse code switches from dot to dash or dash to dot.

It would be fun to play around with this, using the same design idea for other binary trees.

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Langford series

Posted on by John

Notice anything special about the following sequence?

8 6 10 3 1 11 1 3 6 8 12 9 7 10 4 2 5 11 2 4 7 9 5 12

Each of the numbers 1 through 12 appear twice. Between the two 1s there is one number. Between the two 2s there are two numbers. Between the two 3s there are three numbers, etc.

Langford’s problem of order n is to arrange two copies of the integers 1 through n so that there are k numbers between the two ks. This problem has a solution if and only if n is congruent to 0 or 3 mod 4.

You can find much more on Langford’s problem here.

Polish serenity

Posted on by John

Yesterday I ran across the following mashup by Amy Swearer of a Polish proverb and the Serenity Prayer.

Lord, grant me the serenity to accept when it’s no longer my circus,
the courage to control the monkeys that are still mine,
and the wisdom to know the difference.

The proverb is “Nie mój cyrk, nie moje małpy,” literally “Not my circus, not my monkeys”.

Memorizing chemical element symbols

Posted on by John

Here’s something I’ve wondered about before: are there good mnemonics for chemical element symbols?

Some element symbols are based on Latin or German names and seem arbitrary to English speakers, such as K (kalium) for potassium or Fe (ferrum) for iron. However, these elements are very common and so their names and symbols are familiar.

When you take out the elements whose symbols are mnemonic in another language, every element symbol begins with the first letter of the element name. The tricky part is the second letter. For example, does Ra stand for radon or radium?

The following rule of thumb usually holds whenever there is a chemical symbol what corresponds to the first letters of two different elements:

 The lightest/longest-known element wins.

Scientists didn’t wait until the periodic table was complete before assigning symbols, and the easiest names were handed out first. Calcium (20) was assigned Ca, for example, before cadmium (48) and californium (98) were known.

The elements were discovered roughly in order of atomic weight. For example, beryllium (4) was discovered before berkelium (97) and neon (10) was discovered before neptunium (93). So sometimes you can substitute knowledge of chemistry for knowledge of history. [1]

There are instances where the heavier element got to claim the first-two-letter symbol. Usually the heavier element was discovered first. That’s why Ra stands for radium (88) and not radon (86). One glaring exception to this rule is that palladium (Pd) was discovered a century before protactinium (Pa).

Often the element that was discovered first is more familiar, and so you could almost say that when there’s a conflict, the more familiar element wins. For example, Li stands for lithium and not livermorium. This revises our rule of thumb above:

The lightest/longest-known/most familiar element wins.

To return to the question at the top of the post, I’m not aware of a satisfying set of mnemonics for chemical element symbols. But there are some heuristics. Generally the elements that are the lightest, most familiar, and have been known the longest get the simpler names. Maybe you can remember, for example, that berkelium must be Bk because B, Be, and Br were already taken by the time berkelium was discovered.

After using this heuristic, you could apply more brute-force mnemonic techniques for whenever the heuristic doesn’t work. (Whenever it doesn’t work for you: mnemonics are very personal.) For example, you might imagine a registered nurse (an RN) spraying the insecticide Raid on a fish, fish being a Major system encoding of the number 86, the atomic number of radon.

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[1] Chemical elements named after scientists, planets, and laboratories appear toward the end of the table and are recent discoveries.

When was Newton born?

Posted on by John

A young Isaac Newton unwrapping and apple as a Christmas present.

Newton’s birthday was on Christmas when he was born, but now his birthday is not.

When Newton was born, England was still using the Julian calendar, and would continue to use the Julian calendar until 25 years after his death.

On the day of Newton’s birth, his parents would have said the date was December 25, 1642. We would now describe the date as January 4, 1643.

You’ll sometimes see Newton’s birthday written as December 25, 1642 O.S. The “O.S.” stands for “Old Style,” i.e. Julian calendar. Of course the Newton family would not have written O.S. because there was no old style until the new style (i.e. Gregorian calendar) was adopted, just as nobody living in the years before Christ would have written a date as B.C.

In a nutshell, the Julian year was too long, which made it drift out of sync with the astronomical calendar. The Julian year was 365 1/4 days, whereas the Gregorian calendar has 365 97/400 days, which more closely matches the time it takes Earth to orbit the sun. Removing three Leap Days (in centuries not divisible by 400) put the calendar back in sync. When countries adopted the Gregorian calendar, they had to retroactively remove excess Leap Days. That’s why Newton’s birthday got moved up 10 days.

You can read more on the Julian and Gregorian calendars here.

The winter solstice in the northern hemisphere was two days ago: December 21, 2025. And in 1642, using the Gregorian calendar, the solstice was also on December 21. But in England, in 1642, people would have said the solstice occurred on December 11, because the civil calendar was 10 days ahead of the astronomical calendar.