Bill Shillito
@solidangles.bsky.social
Math instructor at Oglethorpe University. Views my own. Talk to me about anything combinatorial game theory related!
He/him. Pronounced SHILL-lit-toe.
Websites: https://www.solidangl.es, https://1dividedby0.com
He/him. Pronounced SHILL-lit-toe.
Websites: https://www.solidangl.es, https://1dividedby0.com
This was the exact moment I learned it's called "maths" across the pond.
October 21, 2025 at 12:00 AM
This was the exact moment I learned it's called "maths" across the pond.
Ah yes, the four basic operations: addition, subtraction, checkmark, and multiplication.
October 11, 2025 at 9:07 PM
Ah yes, the four basic operations: addition, subtraction, checkmark, and multiplication.
UGH, more mistakes that of course I only catch after posting ... here's the corrected graphic.
One of these days I'll manage to post something completely right the first time. 😅
One of these days I'll manage to post something completely right the first time. 😅
October 9, 2025 at 2:31 AM
UGH, more mistakes that of course I only catch after posting ... here's the corrected graphic.
One of these days I'll manage to post something completely right the first time. 😅
One of these days I'll manage to post something completely right the first time. 😅
It turns out we can instead use the quadratic mean, aka the root mean square (RMS).
Squaring makes all the deviations positive, and square rooting at the end gets us back to our original units.
(BTW, here's a video on why squaring really is the best choice:
www.youtube.com/watch?v=q7se...)
[15]
Squaring makes all the deviations positive, and square rooting at the end gets us back to our original units.
(BTW, here's a video on why squaring really is the best choice:
www.youtube.com/watch?v=q7se...)
[15]
October 9, 2025 at 2:23 AM
It turns out we can instead use the quadratic mean, aka the root mean square (RMS).
Squaring makes all the deviations positive, and square rooting at the end gets us back to our original units.
(BTW, here's a video on why squaring really is the best choice:
www.youtube.com/watch?v=q7se...)
[15]
Squaring makes all the deviations positive, and square rooting at the end gets us back to our original units.
(BTW, here's a video on why squaring really is the best choice:
www.youtube.com/watch?v=q7se...)
[15]
Let's look at one more application.
In statistics, we often want to describe not just the center but the spread of a data set.
The first thing most people try is averaging the deviations — but the overs will always cancel out the unders and give zero. So what can we do?
[14]
In statistics, we often want to describe not just the center but the spread of a data set.
The first thing most people try is averaging the deviations — but the overs will always cancel out the unders and give zero. So what can we do?
[14]
October 9, 2025 at 2:23 AM
Let's look at one more application.
In statistics, we often want to describe not just the center but the spread of a data set.
The first thing most people try is averaging the deviations — but the overs will always cancel out the unders and give zero. So what can we do?
[14]
In statistics, we often want to describe not just the center but the spread of a data set.
The first thing most people try is averaging the deviations — but the overs will always cancel out the unders and give zero. So what can we do?
[14]
It turns out these are both examples of a more general idea called an "f-mean." Here's the basic idea:
1. Apply some function f to your data.
2. Find the arithmetic mean.
3. Undo f with the inverse function f⁻¹.
The geometric mean uses f(x) = log x, and the harmonic mean uses f(x) = 1/x.
[13]
1. Apply some function f to your data.
2. Find the arithmetic mean.
3. Undo f with the inverse function f⁻¹.
The geometric mean uses f(x) = log x, and the harmonic mean uses f(x) = 1/x.
[13]
October 9, 2025 at 2:23 AM
It turns out these are both examples of a more general idea called an "f-mean." Here's the basic idea:
1. Apply some function f to your data.
2. Find the arithmetic mean.
3. Undo f with the inverse function f⁻¹.
The geometric mean uses f(x) = log x, and the harmonic mean uses f(x) = 1/x.
[13]
1. Apply some function f to your data.
2. Find the arithmetic mean.
3. Undo f with the inverse function f⁻¹.
The geometric mean uses f(x) = log x, and the harmonic mean uses f(x) = 1/x.
[13]
Again, why did this work?
Well, in physics, you learn that frequency and wavelength are inversely proportional. So can we do as follows:
1. Take the reciprocals of the data.
2. Find the arithmetic mean.
3. Undo the reciprocal with another reciprocal.
That's the harmonic mean.
[10]
Well, in physics, you learn that frequency and wavelength are inversely proportional. So can we do as follows:
1. Take the reciprocals of the data.
2. Find the arithmetic mean.
3. Undo the reciprocal with another reciprocal.
That's the harmonic mean.
[10]
October 9, 2025 at 2:23 AM
Again, why did this work?
Well, in physics, you learn that frequency and wavelength are inversely proportional. So can we do as follows:
1. Take the reciprocals of the data.
2. Find the arithmetic mean.
3. Undo the reciprocal with another reciprocal.
That's the harmonic mean.
[10]
Well, in physics, you learn that frequency and wavelength are inversely proportional. So can we do as follows:
1. Take the reciprocals of the data.
2. Find the arithmetic mean.
3. Undo the reciprocal with another reciprocal.
That's the harmonic mean.
[10]
It turns out we need to use the harmonic mean: flip the string fractions upside-down, add, and divide 2 (the number of values) by the result.
So I have to place my finger to make the vibrating string 2/3 as long.
(BTW, if you lightly touch here, you get what's called a "string harmonic!")
[9]
So I have to place my finger to make the vibrating string 2/3 as long.
(BTW, if you lightly touch here, you get what's called a "string harmonic!")
[9]
October 9, 2025 at 2:23 AM
It turns out we need to use the harmonic mean: flip the string fractions upside-down, add, and divide 2 (the number of values) by the result.
So I have to place my finger to make the vibrating string 2/3 as long.
(BTW, if you lightly touch here, you get what's called a "string harmonic!")
[9]
So I have to place my finger to make the vibrating string 2/3 as long.
(BTW, if you lightly touch here, you get what's called a "string harmonic!")
[9]
When I used to play viola in high school (alto clef represent!) I tuned to A440. An octave up the A string (making it half as long) is A880, and a perfect fifth is E660. (Almost — more on that later.)
Where should I put my finger to play that E?
It's NOT halfway to the octave mark!
[8, lol]
Where should I put my finger to play that E?
It's NOT halfway to the octave mark!
[8, lol]
October 9, 2025 at 2:23 AM
When I used to play viola in high school (alto clef represent!) I tuned to A440. An octave up the A string (making it half as long) is A880, and a perfect fifth is E660. (Almost — more on that later.)
Where should I put my finger to play that E?
It's NOT halfway to the octave mark!
[8, lol]
Where should I put my finger to play that E?
It's NOT halfway to the octave mark!
[8, lol]
The trick is to use logarithms to extract the exponents!
We do a three-step procedure:
1. Take logarithms of the value.
2. Find the arithmetic mean.
3. Undo the logarithm with an exponential function.
What we get is the geometric mean.
(Fun exercise: check this algebraically!)
[5]
We do a three-step procedure:
1. Take logarithms of the value.
2. Find the arithmetic mean.
3. Undo the logarithm with an exponential function.
What we get is the geometric mean.
(Fun exercise: check this algebraically!)
[5]
October 9, 2025 at 2:23 AM
The trick is to use logarithms to extract the exponents!
We do a three-step procedure:
1. Take logarithms of the value.
2. Find the arithmetic mean.
3. Undo the logarithm with an exponential function.
What we get is the geometric mean.
(Fun exercise: check this algebraically!)
[5]
We do a three-step procedure:
1. Take logarithms of the value.
2. Find the arithmetic mean.
3. Undo the logarithm with an exponential function.
What we get is the geometric mean.
(Fun exercise: check this algebraically!)
[5]
We can get a more reasonable estimate with what's called the geometric mean: multiply the values and take a root (here a square root, since there were 2 numbers).
This gives a much better estimate — as you can check on Wikipedia.
But why did this work? How would anyone think to try this?
[3]
This gives a much better estimate — as you can check on Wikipedia.
But why did this work? How would anyone think to try this?
[3]
October 9, 2025 at 2:23 AM
We can get a more reasonable estimate with what's called the geometric mean: multiply the values and take a root (here a square root, since there were 2 numbers).
This gives a much better estimate — as you can check on Wikipedia.
But why did this work? How would anyone think to try this?
[3]
This gives a much better estimate — as you can check on Wikipedia.
But why did this work? How would anyone think to try this?
[3]
Let's start with ol' reliable: the arithmetic mean. The average.
What it does is find the number that's in the "center" of a given data set by "redistributing" the sum evenly across all the values.
But as anyone who's studied triangles knows, there may be multiple kinds of "center!"
[1]
What it does is find the number that's in the "center" of a given data set by "redistributing" the sum evenly across all the values.
But as anyone who's studied triangles knows, there may be multiple kinds of "center!"
[1]
October 9, 2025 at 2:23 AM
Let's start with ol' reliable: the arithmetic mean. The average.
What it does is find the number that's in the "center" of a given data set by "redistributing" the sum evenly across all the values.
But as anyone who's studied triangles knows, there may be multiple kinds of "center!"
[1]
What it does is find the number that's in the "center" of a given data set by "redistributing" the sum evenly across all the values.
But as anyone who's studied triangles knows, there may be multiple kinds of "center!"
[1]
I've been thinking a lot about means recently!
Probably some combination of (1) teaching our data analysis unit in our STEM 101 course at Oglethorpe and (2) all of @howiehua.bsky.social's great posts surrounding Mean Girls Day.
I'd like to show how some of these various means are related.
🧵 [0]
Probably some combination of (1) teaching our data analysis unit in our STEM 101 course at Oglethorpe and (2) all of @howiehua.bsky.social's great posts surrounding Mean Girls Day.
I'd like to show how some of these various means are related.
🧵 [0]
October 9, 2025 at 2:23 AM
I've been thinking a lot about means recently!
Probably some combination of (1) teaching our data analysis unit in our STEM 101 course at Oglethorpe and (2) all of @howiehua.bsky.social's great posts surrounding Mean Girls Day.
I'd like to show how some of these various means are related.
🧵 [0]
Probably some combination of (1) teaching our data analysis unit in our STEM 101 course at Oglethorpe and (2) all of @howiehua.bsky.social's great posts surrounding Mean Girls Day.
I'd like to show how some of these various means are related.
🧵 [0]
At least Italian hasn't caved though! They still use the long scale.
The extra thousand becomes "-ardo" (borrowed from French) at the end of the word but it still is easy enough to understand.
And the term "milione" makes sense: the "-one" suffix makes it "big thousand".
Magnifico!
(5/5)
The extra thousand becomes "-ardo" (borrowed from French) at the end of the word but it still is easy enough to understand.
And the term "milione" makes sense: the "-one" suffix makes it "big thousand".
Magnifico!
(5/5)
September 25, 2025 at 2:21 AM
At least Italian hasn't caved though! They still use the long scale.
The extra thousand becomes "-ardo" (borrowed from French) at the end of the word but it still is easy enough to understand.
And the term "milione" makes sense: the "-one" suffix makes it "big thousand".
Magnifico!
(5/5)
The extra thousand becomes "-ardo" (borrowed from French) at the end of the word but it still is easy enough to understand.
And the term "milione" makes sense: the "-one" suffix makes it "big thousand".
Magnifico!
(5/5)
But wait!
If we use the "long scale" in British English, we get base million, with a sub-base of a thousand, with a sub-sub-base of ten! And everything makes sense!
...or at least that used to be what British English uses. Now they just use the short scale like Americans.
Bummer.
(4/5)
If we use the "long scale" in British English, we get base million, with a sub-base of a thousand, with a sub-sub-base of ten! And everything makes sense!
...or at least that used to be what British English uses. Now they just use the short scale like Americans.
Bummer.
(4/5)
September 25, 2025 at 2:21 AM
But wait!
If we use the "long scale" in British English, we get base million, with a sub-base of a thousand, with a sub-sub-base of ten! And everything makes sense!
...or at least that used to be what British English uses. Now they just use the short scale like Americans.
Bummer.
(4/5)
If we use the "long scale" in British English, we get base million, with a sub-base of a thousand, with a sub-sub-base of ten! And everything makes sense!
...or at least that used to be what British English uses. Now they just use the short scale like Americans.
Bummer.
(4/5)
The new groupings we get are based on powers of a thousand: millions, billions, trillions, and so on.
But isn't it weird how the prefixes don't seem to match up with the powers? It's off by one.
It would be nice if "trillion" were 1000³, but nope, it's actually 1000⁴. Confusing isn't it?
(3/5)
But isn't it weird how the prefixes don't seem to match up with the powers? It's off by one.
It would be nice if "trillion" were 1000³, but nope, it's actually 1000⁴. Confusing isn't it?
(3/5)
September 25, 2025 at 2:21 AM
The new groupings we get are based on powers of a thousand: millions, billions, trillions, and so on.
But isn't it weird how the prefixes don't seem to match up with the powers? It's off by one.
It would be nice if "trillion" were 1000³, but nope, it's actually 1000⁴. Confusing isn't it?
(3/5)
But isn't it weird how the prefixes don't seem to match up with the powers? It's off by one.
It would be nice if "trillion" were 1000³, but nope, it's actually 1000⁴. Confusing isn't it?
(3/5)
For comparison, Japanese is base "man" = ten-thousand (or perhaps we could say base myriad), also with a sub-base of ten.
I remember how difficult it was to learn that "million" is "hyaku-man" (百万), which translates to "one hundred myriad".
I had to completely reframe how I group numbers.
(2/5)
I remember how difficult it was to learn that "million" is "hyaku-man" (百万), which translates to "one hundred myriad".
I had to completely reframe how I group numbers.
(2/5)
September 25, 2025 at 2:21 AM
For comparison, Japanese is base "man" = ten-thousand (or perhaps we could say base myriad), also with a sub-base of ten.
I remember how difficult it was to learn that "million" is "hyaku-man" (百万), which translates to "one hundred myriad".
I had to completely reframe how I group numbers.
(2/5)
I remember how difficult it was to learn that "million" is "hyaku-man" (百万), which translates to "one hundred myriad".
I had to completely reframe how I group numbers.
(2/5)
Hot take:
The English language is not base ten.
It's base thousand.
Sure, we write numbers using the Hindu-Arabic digits 0 through 9, but the way our language is structured groups numbers in powers of a thousand.
If anything, we're sub-base ten.
(1/5)
The English language is not base ten.
It's base thousand.
Sure, we write numbers using the Hindu-Arabic digits 0 through 9, but the way our language is structured groups numbers in powers of a thousand.
If anything, we're sub-base ten.
(1/5)
September 25, 2025 at 2:21 AM
Hot take:
The English language is not base ten.
It's base thousand.
Sure, we write numbers using the Hindu-Arabic digits 0 through 9, but the way our language is structured groups numbers in powers of a thousand.
If anything, we're sub-base ten.
(1/5)
The English language is not base ten.
It's base thousand.
Sure, we write numbers using the Hindu-Arabic digits 0 through 9, but the way our language is structured groups numbers in powers of a thousand.
If anything, we're sub-base ten.
(1/5)
Today's lesson in COR 314!
Students learned how to count in Iñupiaq and write numbers using the Kaktovik numerals (as well as convert between base ten and base twenty). 🧮
Lots of discussion about how the way we're used to thinking isn't the only way to do things! 🙂
Students learned how to count in Iñupiaq and write numbers using the Kaktovik numerals (as well as convert between base ten and base twenty). 🧮
Lots of discussion about how the way we're used to thinking isn't the only way to do things! 🙂
September 3, 2025 at 5:10 PM
Today's lesson in COR 314!
Students learned how to count in Iñupiaq and write numbers using the Kaktovik numerals (as well as convert between base ten and base twenty). 🧮
Lots of discussion about how the way we're used to thinking isn't the only way to do things! 🙂
Students learned how to count in Iñupiaq and write numbers using the Kaktovik numerals (as well as convert between base ten and base twenty). 🧮
Lots of discussion about how the way we're used to thinking isn't the only way to do things! 🙂
Something seems off about the Euler characteristic of this hush puppy.
March 1, 2025 at 12:14 AM
Something seems off about the Euler characteristic of this hush puppy.
Apparently this is now what I do for fun when I need a break from work. 😂
January 16, 2025 at 9:51 PM
Apparently this is now what I do for fun when I need a break from work. 😂
This year's deviled eggs for #NewYearsEve!
Left: Garlic deviled eggs (using toum!)
Right: Ramen deviled eggs (made like onsen eggs but hard boiled)
Left: Garlic deviled eggs (using toum!)
Right: Ramen deviled eggs (made like onsen eggs but hard boiled)
![Left: Garlic deviled eggs (toum [Lebanese garlic sauce], mayo, dijon, green onion, topped with chives and cayenne)
Right: Ramen deviled eggs (miso, mayo, honey, topped with furikake, and the whites have been marinated like onsen eggs)](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:zwo5rlq5oepoq2yx4crus3ma/bafkreicbdukyllgbgpajefngcp4nohgkfovk3oyag5ssdul54qust3rvu4@jpeg)
January 1, 2025 at 2:25 AM
This year's deviled eggs for #NewYearsEve!
Left: Garlic deviled eggs (using toum!)
Right: Ramen deviled eggs (made like onsen eggs but hard boiled)
Left: Garlic deviled eggs (using toum!)
Right: Ramen deviled eggs (made like onsen eggs but hard boiled)
About to take my final for Abstract Algebra (group and field theory).
If this goes well it will be the *last* final I ever have to take.
LET’S GO! 😎
If this goes well it will be the *last* final I ever have to take.
LET’S GO! 😎
December 16, 2024 at 2:28 PM
About to take my final for Abstract Algebra (group and field theory).
If this goes well it will be the *last* final I ever have to take.
LET’S GO! 😎
If this goes well it will be the *last* final I ever have to take.
LET’S GO! 😎