The Mathnificent Word of the Week


Here’s an archive of Mathnificent Words that connect you to where math shows up IRL and convey how mathematical concepts can be a power tool in the workplace, and at home, and in all types of social situations.

They’re presented here in the chronological order that they’ve appeared in the weekly KLM* email.
If you’d like to receive these directly to your InBox, you can subscribe at the bottom of this page.

September 15, 2023: BRAID

Photo by Anthony McKissic on Unsplash 

The beauty of a braid is that it can offer you a new perspective.

The whole time you read this lil diddy about braids, don’t think you need to know what they look like to a mathematician. The image above will work just fine because whether it’s a braid in someone’s hair or a braid being studied in knot theory or physics or cryptography or chemistry or group theory, whatever — the physicality of how it spatially exists is the same.

The super major absolute thing that is required of a mathematical braid is that the place where it starts and the place where it ends is on total lock down. It is fixed. It is anchored. It is moored. Neither the start nor finish of each of the ‘strings’ making up the braid can go anywhere. But in between those two points, those strings can loosen or tighten. They can stretch, they can bend, they can contract (but they can’t deform). Just like a braid in someone’s hair can loosen or tighten in areas of it throughout the day, and depending on what angle you are seeing that braid at different times, you have a changing perspective of it.

But you’re just looking at it. No touching!

A purpose of mathematical braids is that they represent an ability to find, and then observe, and – when they are in groups – compare them to others within that group. But the mathematician doesn’t interact with the braid. Sure, they may rotate it or or rotate themself around the braid, looking at it from different angles – but there’s no interaction. No chance of affecting the start or the end of the braid, because that’s on lockdown.

Today I offer you a braid as a visual representation to inspire the possibility of going out and gaining a new perspective on something that exists and that you have formed an opinion about at some point, in the past.

Go to a sporting event and sit somewhere in the stadium you have never sat before. Have you ever seen a stand-up comic? No? Then, go. Are you curious about a particular place of worship? Go.

Here’s the cool thing: the event is going to have the start and the ending it is going to have no matter if you show up or not — BUT, it might affect you. Hopefully in a good way, possibly in a bad way, who knows. Just remember, here’s the super major absolute thing that is required of you: Don’t intercede. Don’t interrupt. Don’t engage or participate. Just take it all in.

This is not about going to that bookclub that you have missed for a few months. Or that poker game. Or that monthly happy hour or weekly status meeting. This is all about total observation from a position that you haven’t observed this particular thing before. You don’t want to become the cause of a tangle happening in that braid (which is totally a math thing! Tangles don’t just happen in yer hair, they happen in mathematical braids in a very similar way to a braid in yer hair! I know. I know)!

Let us know if you gain some new perspective that maybe gives you a little reset at this time of year. Then shout it out on the socials 😘

August 29, 2023: LUNE

Photo by Jordan Steranka on Unsplash

Go talk to the moon.

95%+ of the time you look up at the moon, you’re seeing a lune shape. Many hundreds of years ago, mathematicians were way into using algebra to ‘square’ shapes that weren’t, well, originally square. Triangles, circles, polygons, you name it – they wanted to square them. And Hippocrates of Chios was the first to do this with a lune back in 440 B.C.

Now, I’d like to come clean to you and say that I don’t know why these mathematicians were compelled to do this squaring thing. I’m sure there was a very good reason that is now used in all sorts of engineering feats – and if you do know why they did this, please do let me know!

But I decided to assign the lune as the Mathnificent WoW because I wanted a natural segue into alerting you about the appearance of the Blue Supermoon that will show itself tomorrow, August 30th.

First off, a blue moon happens about every 2.5 to 3 years since a moon’s cycle is 30days so, using the Gregorian calendar of modern times, it takes that amount of rotation time to fit a full moon in the first & last couple of days of that month.

Secondly, this particular blue moon will be at a relatively close distance to our planet and, therefore, will appear to be quite large. As noted in the most recent KLM* essay, the orbit of the moon is not circular – it’s a superellipse, more oval-like – so the frequency of this closeness of the moon to our planet is a bit irregular (compared to the regularity of a blue moon).

All of this is to say that the appearance of a Blue Supermoon is infrequent; the next one will occur in January 2037.

Apparently the month of August has been – for moi – abundant with geometry, with all my writing of planets and shapes, superpositions and superellipses, and the math that underlines alllll of it. I say, channel all that math into the cosmos, take this as some sort of (super?) alignment of power, and go talk to the moon.

August 9, 2023: OR

The choice of this, or that, or both?

We use the word or all the time. I’m sure you use it multiple times a day, everyday. Right there in our decision-making, at the bifurcation point where the road splits in two directions and there’s a call for a choice to be made. And so many times, those two points are headed in opposite directions from each other. Make a choice to go in one direction and there’s that lil itch that will only get scratched when you receive some confirmation – verbally or visually or maybe even viscerally – that you made the ‘better’ choice. But what if your better isn’t the better for everyone else on your ride?

In ye ol’ math world, the word or is used as an inclusive word.

When the word or is used in a mathematical problem, it’s a connector word, it’s actually an operator [oh hi Mathnificent WoW from April 25th], it’s a “mapping between two function spaces”.

P ∨ Q

This statement is true if:
P and Q are true,
but it’s also true if just P is true.
And it’s also true if just Q is true.

Situationally, the math definition of the word or can be an “always looking on the bright side” sort of word; it’s A Brightsider 😎. I mean, in math-speak, the likelihood that you’re gonna be able to say that a choice between two things is “true” is 3:1, right? I like them odds.

Let’s say you wanted to see the ocean today and you thought that when you had to choose which direction to go in, you had made the ‘better’ choice. And though you eventually found out you had made what wasn’t the better choice for you and your goal, did you then take a moment to consider that your choice actually was good for someone else in your vehicle? Did they get to see something they wouldn’t have ever seen or experienced if you had gone the other way? Could you brightside it to alleviate some of that pressure you have put on yourself to always make the better decision?

Sure, your ‘not-better’ decision may have absolutely caused the need to make a u-turn to get back on the original path, but unless this was a life/death decision, there’s probably somewhere in there you can brightside it – even if only for a moment. It might just be a lesson learned sort of thing.

Or, not.

July 25, 2023: WAVE

Photo by Li Yang on Unsplash

The most important sum sum summertime calculation of them all.

When you’re about to interact with a body of water, there’s a few mathematical components of a wave that you (un)consciously take into consideration:

  • Its amplitude, the measurement of vibration and spin that’s contributing to its arc.
  • Its frequency, the positioning of the waves as they cross a particular point.
  • Its wavelength, the expanse of which contributes to its overall momentum.
  • And its phase, the timing & pacing of just one revolution of a wave, which leads us to that one thing any practiced ocean swimmer realizes is the most significant component to keep an eye on.

You’re looking for the shift.

Yup, the shift in the phase. It’s the measurement of the always changing timing between the waves, and oh yes, I know you know that sometimes that shift can be hard to handle, but, oh, when it is just right for you, it’s great.

Sometimes it is just right when it’s low and steady and rocking you like a baby to sleep as you sit in your rowboat to fish.

Sometimes it is just right when the wave phase is extreme because it’s perfect for surfing yet the shift is lonnnng so that you can get back out past the break for your next ride in.

And sometimes it is just right so that you can jump the waves, feel safe in the big water with friends, and cool off on a stupid hot day.

There’s plenty of hawt summer playtime left. The shift will invariably, well, keep shifting. BUT, here’s what’s the coolest part of this particular hotness: If you’re unable to lock into the shift that is just right for you when you’re needing it to be a particular way?? It’s guaranteed to change. Your ‘just right’ may be on the horizon.


July 4, 2023: EXPRESSION

As the great American poet, Madonna, once said: Express Yourself

Have I ever mentioned to you how I have absolutely not-at-all-any-sort-of-poker-face? I am incredibly expressive. It’s a gift, it’s a curse, just depends on the situation I find myself in.

The myriad of expressions I, errrr, express – with my face, but also in my gestures – are a representation of how I am feeling in the moment. Of course, sometimes you don’t want to be such an open book, not just in a poker game but also in a negotiation or when someone else is communicating their feelings towards you and perhaps you aren’t quite at a point of mirroring those same feelings back to them. 😕

You may assume that expressions are the same thing as equations. Nope! A key difference is that expressions are one-sided and equations are two-sided.

3x + 4(x – y)

3x + 4(x – y) = 4(x + y) – 1

It appears that a mathematical expression has a lot in common with my expressive non-poker face.

In your world and mine, the value that any expression contributes to any sort of exchange of information IRL – be it a poker game, a negotiation, or an emotional reveal – is just one of the sides of that conversation – aka, the equation.

An equation, on the other hand, is gonna tell you the whole story. It’ll include an expression and it will also include some sort of equivalence to that expression, right there on the other side of the = sign.

Today is Independence Day here in the U.S. Put a flag up, whatever flag you like, that you feel best expresses you. Acts of expression can be revealing, passionate, and even empowering. Consider, though, that they are also the representation of just one part / one side of a person’s own story / equation.

June 20, 2023: CONGRUENCE

Photo by Tim Chow on Unsplash

There’s time for thought when you pause for a movement.

Last week’s essay on equality (vs equivalence) took me on a whole ride of the mathematical definitions of the word equal – and absolute value, and similar, and modulus(!), and congruent – and that was just in the discipline of algebra.

In the discipline of geometry, I encountered the word congruence again, but now with the words ‘…taken to the other by a motion’ — and I knew immediately it was gonna be a Math WoW.

Words used in geometry offer so many opportunities to humanize math in a relatable way, and I think that’s because there’s a physical component to shapes and figures that easily lend themselves to visualization. Here, congruence is all about the realities of considering different perspectives of observation through movement that’s mandatory, all to be completed before you can make a conclusion about the relational equivalence of the figures before you.

There’s an inferred pause in our action when we take a moment to consider something. The optimist in me also hopes that when we take the time to pause to evaluate something as significant as whether or not a set of figures – or objects, or perhaps humans – are equivalently valued, our inherent biases have the time to show up, be seen, and be acknowledged.

Congruence is more than just looking similar; it is equivalence in the inner breadth as well as in the outer appearance.

June 6, 2023: ELIMINATION

Get it gone. Clear it out. Maybe even make things easier.

The Mathnificent WoW on March 7th was substitution. At first blush, the action of substitution seems to be used in a very similar way to this method of elimination, but just as the common definitions of the words substitute and eliminate are not the same, their mathematical actions are used differently, too. While the former is about repositioning the elements so that their values are redistributed across two equations that are being solved for, the latter is outright ixnay’ing that element – that variable – outta there, with the overall goal of making a process or a system easier and, ultimately, solvable.

We are in the 💚 of spring cleaning season right now, a tradition dating back to biblical times and across many cultures. Now’s the time to not only eliminate that which is of no use to you anymore, but to eliminate in order to clear out the space.

In linear algebra, the elimination method is specifically used when you’re trying to solve two equations simultaneously. The thing to determine before you kick that variable outta there is understanding that variable’s value in order for you to determine if – by sweeping it out the door – you’ll have made the overall process or system easier (in terms of coming to a conclusion, an end). I think this is a great way to consider using it IRL, too.

So, it’s not about cleaning out that one junk drawer in the kitchen. I mean, go ahead, do that, it’s a great feeling of accomplishment, for sures. This elimination would have a broader reach. As in – this type of elimination is more about kicking a habit that not only gets your health back on track but may also solve some other things, like perhaps some not great relationship patterns.

One stone, two birds – or maybe even all the birds! Hrrrmm, I think I want to get that ol’ idiom out of my lexicon. I really don’t have a problem with birds.

May 23, 2023: POINT

Points are one of the most fundamental objects
in all of mathematics.

Writing about symmetries and spacetime last week put me in a dimensional state of mind – 2D and 3D, and then 4D started blowing my mind and not in a way that felt particularly good, so — I backtracked out all the way back to the base of it all, to the point.

Visually, a point is usually represented by a dot. The basic structures of higher dimensional geometry – the line, plane, space, hyperspace – are all built up of infinite numbers of points arranged in a bunch of different, though particular, ways.

A point has no depth or any length — the only thing it has is its defined position. If two points are connected by a series of points — aka a line — then it follows that two positions have then been connected. A point marks the beginning – of a line, of a shape, of…well, anything and everything.

You likely use the word point as a synonym for purpose. Or for an exact moment in time. Or just using your finger to emphasize a particular direction or item.

In that mathematical definition, it’s so simplified. A point is solely about its position – and then I suppose there’s an assumption that it will, or it at least could be more. Additionally – and this really got me – a point cannot be defined in terms of any other points that you may have come across before in your day-to-day experiences of, well, other points. It’s its own thing, my dude.

There are soooo many definitions for the word point; I counted more than 20. Though I don’t like oversimplifying things, I do like to break things down to their most basic state in order to get a new perspective — maybe even a perspective that was the original intention.

With a 3day holiday weekend coming up here in the states as Memorial Day is celebrated, I feel like we’re all pushing to get projects done before the summer – and the summer mindset – begins.

So – I’m thinking, what’s my point this week? Meaning, what’s the position in space that I’m wanting to define here for myself (and myself only) right now, this week – especially if I sense that I’m pushing? Is it that I want to feel like I completed something? I kinda want my point to be uncrowded, to be clean and tidy, actually. I like the grounding in this – a lot more than having to define a point as a purpose or as a reason. I like the physicality of locking in that point as just a place – a position – in space, as is.

Let me end this one with a good ol’ math pun 🤓
Without geometry, life is pointless.

May 9, 2023: LOCUS

There’s an underlying issue if you can’t even reach minimal satisfaction.

Pardon me, but I’m wondering when was the last time that you checked your locus? You’ll find it at your work location, or your home location, or some other location where you’ve previously defined how you want that physical/spatial place to exist in order to just meet your needs, at a minimum.

In the definition of locus, the word ‘satisfy’ plays prominently and in mathematics, that word means the same thing that it means in our non-math daily lives; when something – or someone – is satisfied, then a need has been fulfilled, the qualifications have been met, the information is adequate. There’s nothing to complain about, and to be fair — if I have what could be called an urgent need (anything from health to hunger aka h’angry to horny) — having those needs satisfied is exactly what is wanted and expected. That’s it. No more no less.

The way to check your locus in relation to a location is to focus on & honor whatever set-of-all-points of a condition that you thought of the first time you evaluated that space for your needs.

It’s those “I can make this space work for me” thoughts like:

  • With a new coat of paint, this room will be ideal.
  • Having an ergonomic chair will make all the difference.
  • As long as proper window coverings are installed, this space is workable for everyone.

Here’s my own example: Right now, there’s a bunch of work being done on my house, work that we said we’d do when we bought the house…15 years ago. Rooms painted that we’d never painted. Floors refinished in the way that we instinctually knew would make the entire place feel more comfortable, more cozy. And a proper desk space for me to be able to do research & write at – instead of at the kitchen table or sitting on the couch.

Ooof, I wish we had done this years ago. I’m legitimately in a better place, physically and emotionally. I had allowed workarounds to become permanent. I had created reasons to make those workarounds acceptable. And now I’m wondering, what if I had honored that set-of-all-points earlier, never mind 15 years ago, but what about 10 or even 5 years ago?

If you realize that that set-of-all-points actually can not be satisfied, like not ever, be it by you or by other powers that be – then is that location itself even satisfactory? It could be time to move to a new space.

Or maybe it’s time to change the condition that needs to be satisfied in that space, by the set-of-all-[those]-points. If that room in your house was supposed to be your office yet it’s also – still – a (frequently used) guest room, do you honor yourself by getting a desk at a shared work space outside the home? The locus itself would be the same except for the designated location it’d exist within.

Or could you build-out your garage as a guest space, and then you could rent it online for short-term rental for travelers (which could then subsidize some of that shared workplace rent)?

Honor your locus. You defined it for a reason in the first place.

April 25, 2023: OPERATOR

I said what I said.

At first blush, as I broke down the math meaning of operator and compared it to the common everyday meaning, the definitions seemed to be the same across the board; “a mapping of two spaces” totally jives with “someone who performs the practical work at a telephone switchboard.” Two spaces / two entities / two people are being mapped – they’re being connected – by an operator.

But then – oh yeah, it happened again – that mathnificent bit of the word operator called out to be seen (it’s not gonna be ignored, Dan! 🐇😉 )

That mathnificent bit is that the output of those two function spaces – that output of a message, or a direction, or some other form of communication that the operator will use to connect one person to the other – that output must have a specific quality. The math goddesses dictate that the quality of that output must be well-defined and in ye ol’ math world that means it must have the quality of being unambiguous and uncontradictory.

That message can not be vague. That information is not in opposition nor is it inconsistent. This is a fantastic distinction.

If I order a pizza, and the order is communicated to the person making the pizza via an order app or by the person who answers the phone and writes it down, I want the exact pizza that I ordered; if I ask for extra cheese, I expect extra cheese.

If, as I leave my office to go see a client, one of my co-workers asks me to communicate that they’ll be sending that client an important email which must be replied to by a specific date and time, and then I tell the client that they’ll be receiving an email from my co-worker which they should respond to when they have a free moment — I completed the operation that my co-worker asked me to do, right? My messaging was definitely not vague — yet, it was also not correct. It’s actually contradictory to what I was asked to communicate. I’m such a jerk.

So, here is my wish for you: May your future be filled with operators that aren’t jerks.

April 4, 2023: AFFINE

I think you’re gonna be more than (af)fine with affine.

You don’t need to read the most recent KLM* essay on Self-Similarity to get the full shah-POW🌟 of this MWFYW but, dang, I didn’t know about the math-world definition of affine or affinity which then leads to self-affinity till I was doing my research for that essay cuz, wow — they are totally related.

In geometry, according to Mandelbrot (our grandaddy of fractals), self-similarity is the carbon-copy action of “shapes scaling by the same amount in every direction.”

Affine – and onward to affinity then to self-affinity – is just like self-similarity but the carbon-copy action has those shapes scaling in different amounts in only some of its directions. Mandelbrot referred to self-affinity as turbulence.

Self-affinity is turbulent self-similarity.

Now, take my hand (no need to squeeze that hard, sweetie, we’re just talking about mathematics) and let’s look at what’s probably a more familiar usage of this Math WoW for you, as exemplified in this sentence, “I have such an affinity for confident nerds.” Here, affinity means having a particular liking for something or, more often, for a person or people. We commonly use the word affinity in a relational context.

Just as applying self-similarity to real life uncovers how long term processes & systems can arise biases within, self-affinity may be just the thing – that good version of self-similarity – that I was looking for.

A group that has self-affinity rather than self-similarity would be one that has experienced turbulence within as it does that carbon-copy action of scaling. As ‘they’ say 😉, whether you’re part of a couple or a group of people – long-term healthy relationships have disagreements; that’s definitely a type of turbulence.

Groups transform over time – sometimes scaling up and sometimes scaling down. With self-affinity, if you’re realistic about wanting to ensure long term sustainability of that relationship or group, that scaling actually doesn’t – and shouldn’t – scale equally every step of the way.

Huh. Where have you been all my life, affine?

March 21, 2023: PARITY

The quest for attaining parity can feel like
you’re on the search party for a chameleon.

The common definition of parity is the state or condition of being equal in terms of numbers and proportions. For example, a gender parity would be the balanced (in terms of numbers/proportions/ratio) representation of all gender identities within a group or organization – most ideally in leadership. Though some folks may use the word to also mean equality of power or influence, that’s technically not correct. Parity, no matter what context it’s used in, is a descriptive word only; it’s not in relation to value or policy.

But now, here’s the math-world definition(s) of parity that is oh-so-close to meaning the same thing yet… nope… it’s different enough that I’d like us to take this math-world definition(s) into consideration when we’re working on something IRL where reaching parity is the goal to be achieved.

This definitive difference is a bit of an eye-opener, truth be told. The overall takeaway with either of the math definitions clearly states that parity is a binary descriptor, an either/or. Since I don’t ascribe to the archaic definition of gender being binary, then the phrase gender parity is a bit of a head scratch, no?

So goes what I’m asking you to consider: In addition to a balance in terms of proportions or ratio, howz about a balance of the attributes of negative / positive & an equivalence of left / right? Y’all get to decide the most fitting way to define the words negative, positive, left, and right, based on the conditions and structure you are doing your parity work within.

I’ll admit, sometimes it’s like being a part of a dance, kind of like the hokey-pokey, with a barker calling out do this, then that, now this — and that’s what it’s all about.

I still go with the overall goal being to bring everyone together so that there’s a representative blend into one cohesive unit. Though your quest for parity may create a flicker of standing-out at first, as the mixing continues the hope is that each new addition will blend in with its surroundings and attain a balance of that negative / positive, left / right.

sidenote: I am now drawn to writing any essay named…what else… The Parity Trap
Thanks everyone, please tip your bartenders on your way out.

March 7, 2023: SUBSTITUTION

I’m wondering if it’s time to move around the support players
in order to simplify the flow of the entire game.

Allrighty. We’re ten weeks into the new year, how’s 2023 going for ya?

Though I’m pretty self-reliant, I absolutely need the support of my peers to reach my goals for expanding the audience of Kate Loves Math*. We gotta keep on lifting each other up, right?

Now, I’m a fool for baseball and a couple of weeks ago Major League Baseball spring training began. Spring Training is like a dress rehearsal; everyone’s in uniform and they’re going through the entire 9 innings and they are playing to win, but each team is also moving players around and testing them in different positions and changing up batting orders with a lot more frequency than they do during the regular season. And there seems to be a pitcher substitution almost every inning.

Which got me thinking about the past 9 weeks (oooh, like 9 innings ⚾️) and whether it might be a good idea to just, you know, evaluate my own (support) team and – if needed – apply the mathematical process of solving by substitution.

Here's a mathematical example of solving by substitution:
You have two equations to solve where 'x' and 'y' are each representing a number.
                          x = 2y - 4      and     2x = 3y - 5
Since the value of x and y are the same for both equations, if you can figure out what number x represents in one equation then you can substitute that same number for x in the other equation. This should help in figuring out the corresponding y value and solving both equations.

Quite simply, I’m reassessing what I knew about my peers in the beginning of the year based on recent conversations. There’s definitely some people (let’s call them ‘x’) who have additional knowledge and skills that I didn’t know – 9 weeks ago – that they possessed. So, I’m wondering if I can substitute the specific skills someone has to support me in reaching one goal for another goal that I hadn’t even considered they’d be able to support me with.

And that all made me wonder if you might want to try this solve by substitution process yourself. Sometimes while working a plan there is new information about a key element that’s casually served up, yours for the taking – and for the re-positioning. Lemme know if you run through some scenarios of your own and if this analytic process helps you – and your team – better accomplish what you’re going for.

For those who actually read those equations above >>>

February 21, 2023: PERMUTATION

It isn’t so much about taking a linear approach,
it’s about considering a thoughtful approach.

When you need to do – or lead (or be a part of) – something that has multiple steps from start to an ideal-looking finish, to just get the motor-running on the process you may do the easiest thing first and save the “hardest” thing for later down the line. Or, I’ve known some people to tackle that hardest thing first so the rest of the process is a relatively easy downhill slide.

Though the word permutation is commonly defined as “one of several possible variations that something can be ordered or arranged” – please allow me to introduce the mathematical definition of the word, which actually dictates that there is a specified order to a specified number of items & actions that must be followed in order to get to the finish line that is desired.

You see, in ye ol math world, when the order of the steps in your process don’t matter, then that’s simply known as a combination. Thankfully, there are probably more combinations in our lives than permutations. 🤞🏼

The IRL mathematical permutations I am thinking about are the ones that revolve around human-centered actions. I feel like I’m seeing a lot of them in the news lately, like swaths of layoffs at large companies or the mass eradication of homeless encampments.

Employing the concept of applying a mathematical permutation to the order of the steps in your process may absolutely – in fact, it’s quite likely – add additional time to the process. Sure, those layoffs may legit need to happen and need to happen ASAP and you want to time it ‘right’ with the next payroll cycle, but don’t do layoffs on a Friday, that’s f’d up for everyone involved and assuming a weekend of time away from the office will then make for a calmer “fresh start” on Monday…nah, that just ignites more questions.

Or, making executive decisions to satisfy the loudest $$$-controlling complaints in your ear yet not taking the time – not insisting on taking the time – to talk with the homeless population that’s directly affected by the actions of their only safe shelter being destroyed, that, too, is f’d up.

Thoughtfulness and respect for our fellow humans is the social currency that doesn’t ebb along with the rate of inflation. This is what is meant by human-centered design.

So here’s what I ask of you: the next time you find yourself about to start something that will directly affect the lives of other humans, I hope that instead of just evaluating the impact on your life, you’ll decide to evaluate thoughtfully the impact you’ll make on others as well. If your process needs to now add or reorder some steps, so be it. Then forevermore you may now have, and insist it will always be, a permutation of that process.

February 7, 2023: ARGUMENT

Here’s why the phrase let’s agree to disagree is the most asinine and unproductive thing to say.

In non-math saturated lives, the word argument is usually seen to have negative connotations. If you tell someone you had an argument with someone else, the response will be something like, “Oh, that’s a shame. Are you all right?” or “Who won?”

In the world of mathematics, arguments have a necessary presence in just about every function (equation, problem, challenge) because they are the very definition of the value of the function (equation, problem, challenge) at-hand in order for everyone to be able to move towards a successful joint outcome.

If each mathematician assigned their own personal value definition to an argument, it’s kinda obvious that – when working with others on that function (equation, problem, challenge) – the likelihood that the solution would be a success is somewhere around nil (aka zero).

So, why do we think we’ll find true success on a team if there’s a couple folks doing that whole let’s agree to disagree (so we can all just move on) passive-aggressive dance?

Additionally, if you read last week’s essay about variables, then you’ll understand that since an argument is also a variable, that means there’s an acknowledgement & awareness that’s required to be considered about those conditions where that argument is existing within, in order to accurately determine its value.

Here’s what we can learn from the mathematical definition of an argument:

  • Two or more people are willingly participating in this exchange of information. Less than that and you’re witnessing a rant, not an argument.
  • The topic of the argument remains focused on and applicable to the conditions of the function (equation, problem, challenge).
  • The argument is complete when all participants actively agree on the value that’s been defined for the function (equation, problem, challenge). Ergo, agree to disagree does not qualify in this context.
  • Respectful communication abound. Duh.

I think arguments have always been meant to be a positive way to move through a disagreement. Because when everyone gets to the other side of that argument, don’t you physically feel lighter?

January 24, 2023: MARTINGALE

What a lovely sounding word for something that has everything to do with risk.

The original usage of the word was with gambling to describe a betting system where when a bet was lost, one would double-down on their next bet, continuing that method until they won. The assumption was that probability dictated that, eventually, they’d have to win at some point. Of course, that was fruitful if and only if all other things were equal – and the gambler had infinite $$$ to bet with. There’s also a variation that, when a win does happen, the next bet is half of the prior, and the betting continues till the gambler breaks even.

The Martingale System was adopted (and adapted) by early 20th century mathematicians who worked in Probability Theory; it’s their definition that I’m referencing today. It also clearly calls out a randomness factor from start to finish.

Yesterday I realized that I am already assessing my own 2023 goal progress (yeah, just a few weeks into the new year 🤪) and since I’m such a fiend about planning out each and every step in a plan and looking for patterns to success, the unexpected things – good or bad – rarely happen. This stirred up the memory of what someone said to lil ol’ risk-adverse me a handful of years ago.

“If you keep trying to study the risk out of risk, you’re going to miss an opportunity.”

Dang it.

So I offer to each of you the consideration to, occasionally, put on your Martingale Glasses™️ – especially when you might be about to immerse yourself into a time-suck of investigating the path to a guaranteed outcome. Take a step (or two or three) in an unknown direction that has a chance to reward you with something you hadn’t known of and that you’d deem to be as equally good – or maybe even greater – than the direction you’re most familiar with.

I know you’re smart so always check your gut first, know your personal risk boundaries, and be aware of all who may be affected by any decisions you’d make. But, hey, sometimes the law of probability is bound to go in your favor, right?!?

January 10, 2023 NEIGHBORHOOD

One of the most important things in mathematics are relationships.

The other day I was thinking about manifolds because, c’mon, look at this wacky shape, isn’t it cool? And while I was reading about them, I was reminded that just like most things in the study of topology (which lives within the massive discipline of geometry), defining what the neighborhood is within a manifold is integral to the very definition of the manifold itself.

Now, I realize what I’m about to posit sounds like a Carrie Bradshaw question, but: Is defining what my neighborhood is an integral part of defining who I am?

For my computer science friends and any other readers who have dabbled in the world of information technology, I’d bet you’re quite familiar with neighborhoods in regards to data networks. These tech networks also have trees (that have branches!) and paths (of information going from Point A to Point B and on and on to other Points). It’s all further evidence that people have been humanizing math forever-eva. All of this was born from stuff like graph theory, and combinatorics, in addition to the aforementioned topology and geometry.

So, in my process of defining my neighborhood using the mathematical definition, I label myself as the ‘given point’. Then I need to specify the value of the distance from myself to all those points in my set (AKA in my neighborhood). Do I take on a value that is large, as in: Is my neighborhood global? Or do I define the measurement of my neighborhood to actually be local, so maybe it’s synonymous with the borders of my state, or my town, or the company I work for. 

Looking at the word value and considering how important that word is to me IRL, I realize that since we’re now talking about human relationships, I need to recognize one of the other meanings of the word value: worth. Not the worth of others in the neighborhood nor the worth of myself, but the worth of the relationship itself; the worth of generating it and maintaining it, for everyone that is a part of it.

And that’s what convinced me that defining my neighborhood does indeed contribute to defining who I am; it’s kind of the mathy version of Show me who your friends are and I’ll tell you who you are, except instead of focusing on the reflective part of that saying, the focus is on – as with any good mathematical concept – the relation of all the elements that are present, and what they can produce together, being the parts of a whole (neighborhood).

 In 2023 and beyond, my neighborhood includes all y’all. The value (measurement) I specified is my potential reach to readers like you as well as the value (worth) of the relationship we have with each other.

How do you define your neighborhood, especially as you start putting action to your 2023 plans?

December 20, 2022: PROOF

“Math is a search for objective truths, while proofs are the search for subjective agreement.”

That quote is from mathematician Jay Cummings from his long-form mathematics textbook: PROOF. I think it’s a succinct observation of how math is – just like all things are, for all of us, everywhere in all our lives – always looking for and sharing affirmations to statements that we can then, collectively, move forward with as Truth (with a capital ‘T’).

I do love that the word proof is one of those words that has multiple variations on its meaning, depending on what it’s being associated with. Are we using it in mathematics? Referencing alcohol? Is it a typeset for approval? Or, in adjective form, is it to resist or repel (waterproof)? Or how about as a verb when you’re baking and using yeast?

In all the possible definitions of the word, there’s a consistent theme, and that is that there’s a definitive and absolute certainty about it that can be verified by others. And that’s why – instead of making a resolution or a goals list for 2023 – howz about making a proof? A Personal Proof.

Here’s the most basic of the basics of a proper Direct Proof:


  • logical and crystal clear phrasing.
  • formality in its language.
  • brevity in its length (keep it under 10 sentences).
  • understandable everyday language.
  • familiar mathematical phrasing, like, “If {abc}…then {xyz}, as well as phrases like “If and only if.” It is straight forward.


  • any run-on descriptive sentences.
  • trivial information. Every word in every sentence has a reason to be there.
  • any statements that could be open to personal interpretation by the reader

NOTE: If you are writing your Personal Proof for ‘future you’, write down what you’d consider to be the goal date for it to be True since a proof is always written in the present tense.

You start with a theorem — your proposition of what is absolutely true.
For example, you might say:
∀ I can lift 100lbs of machine weights above my head at any U.S. compliant health & fitness facility.

You follow with a statement(s) (aka a lemma) that, in consecutive statements, supports that theorem:
If I go to my neighborhood gym, then I can lift 100lbs on the hammer strength machine.
My neighborhood gym is a U.S. compliant health & fitness facility.
∴ I can lift 100lbs of machine weights above my head at any U.S. compliant health & fitness facility.

Keep in mind that your theorem does not need to be an obvious statement; go ahead, surprise other people in your life, show them how you have a thought-out logical and understandable plan-of-action that you’ll be able to say, at a pre-determined date, is the Truth.

I like the idea of writing a Personal Proof because it is a concise statement that states a commitment you’re making to yourself. If your yearly goal lists tend to have general nebulous statements, make a few Personal Proofs; it requires you to focus and be realistic about what is needed in order for you to embody your Truth with a capital T.

December 6, 2022: REFORMULATE

There’s a humbleness in it that shows up as a strength in character.

The root word form is in many mathcentric words: formula, formal, formality, formalism, formulate and of course simply the word form in and of itself. It suggests a structure of some permanency, something that is built in a definitive way, and of course when you’re talking about math or science, there’s likely deep thought behind the action.

You mathematically reformulate something when the new version:

  • Won’t look that different to those on the periphery; there’s minimal ripple effect
  • Will still share the same properties from its previous version
  • Will be more efficient in its process
  • Will bring a sense of ease to all that interact with it
  • Will openly reveal what wasn’t working in the prior version

What I like about this MWFYW is that the reformulation is a change that is unequivocally for the better – where ‘better’ is about efficiency and ease.

It can be hard for a mathematician to have to reformulate. They will kick and maybe even scream if you tell them that their only path to success is – after working on the same thing and being on the same path for days months years, something that they’ve modeled, that they’ve tested, that they’ve FORMulated, dang it, – that they need to reformulate because there is something very apparent blocking the path to success.

I can also tell you what reformulating is not — especially if you’re considering a reformulation in the workplace. Reformulating ≠ pivot. When businesses consider pivoting, they’re wanting to alter to some degree their core service or product in order to better meet the needs of their clientele. Pivoting is looking externally to dictate the alteration, and therefore the impact to internal is actually a reaction.

But when you reformulate, you are looking inward, first and foremost, to make your alteration. I think when a business reformulates, they’re self-aware, they’re saying “it’s me, not you.”

I believe reformulating doesn’t need to be dramatic, but it can be personal. Your business might finally accept that a new position needs to be created for responsibilities that you can’t manage yourself anymore; it’s a betterment in efficiency for everyone. A reformulation in your workplace can alternatively be automating something that doesn’t require a human component for smoothing out a process for ultimate success.

And for yourself – well, consider reformulating instead of starting over. Reformulate instead of pivoting. Reformulate instead of reacting to an outside causation that you interpret as meaning that you gotta scrap it all and do something else.

Honor your original formula, pay attention to the internal workings (that work) and make alterations for efficiency and ease.

November 22, 2022: BIFURCATION

The knowledge we gain from our experiences informs us more than you may realize.

According to the folx at Merriam-Webster, the common definition of ‘bifurcation’ is: the point or area at which something divides into two branches or parts. It’s a handy word to illustrate when one road splits and becomes two roads, or when a tree grows a new branch from an existing branch.

Then there’s how the word ‘bifurcation’ is used in the mathematical universe. The term ‘bifurcation’ has more details than what is here in the MWFYW. There’s a whole thing called Bifurcation Theory, and there’s even three different categories of bifurcations, and I know you don’t want to shimmy down this rabbit hole right now. But I do want to highlight what hooks into me about bifurcation.

A bifurcation starts because of a small change in a parameter. A parameter for you and me are the conditions that we exist in. And, you know, small changes in our conditions happen all the time. Sometimes you work late so you end up catching a different bus. Or sometimes there’s a sudden rainstorm and your feet get soaked. Different experiences can happen due to a small change in our conditions.

Also, a bifurcation is all about a qualitative change. We’re not talking about a quantitative change, this isn’t a numbers or statistics sort of moment. This is a change that happens at the crossroad moments in your life that is based on the human side of things, on feelings and on memories and – good or bad – it’s based on a reaction to something that was experienced by You.

So, you’ve got a small change that actually triggers the bifurcation point to appear in the first place. And then the experience from that small change also informs which of the two paths should be chosen at that decision-making moment. I guess that’s why some mathematicians call the bifurcation point a “fork of possibilities.”

To me, this is a reminder that when I come upon a significant decision-making moment in my life, I don’t just check my gut, but I trust my gut because I know that it’s my experiences that got me to this fork of possibilities, and my experiences contain the truest of information to help me choose which path to take.

November 8, 2022: CYCLIC

As humans, some of our patterns of behavior evolve, while other patterns just repeat and repeat and repeat.

As I’ve mentioned before, you are a brilliant person, so seeing this Mathnificent Word For Your Week (MWFYW) I’m sure you immediately summarized that the first three letters of the word (cyc) are at the root of a ton of other words you’re already familiar with — words like cyclone, or cycloid, or cyclical.

Though all those words have different definitions, they all draw the base of their definition from ‘cyc’, meaning “some sort of recurrent movement.” That movement is implied in botany’s definition of cyclic, “arranged as parts of a flower,” in chemistry it’s “relating to a compound that contains a closed chain or ring of atoms,” and in the mathematical branch of geometry it’s specific to “having all its vertices lying on a circle.”

I focused on the algebraic definition, where in the photo the pinwheel rotates each of its four elements – equal in power – in a recurrent movement so that the first element immediately follows the last, and the pattern repeats and repeats, never evolving in any way; it’s just sorta…there, doing it’s thing, probably bringing a lil joy to all who happen upon it. It’s fine.

When I first came across this MWFYW, it got me to realize that, huh, I was in a cyclic way of living. And it really was more than just cyclical, it was algebraically cyclic. I had given equal importance – equal power – to each of the elements in my life, in repetition.

The big thing (that I totally realized but had also pushed to the back of my brain so I could ignore it) was that I really didn’t feel like I was on the proverbial hamster wheel, going through the motions. Nope. I was fulfilled — while also being unfulfilled. I was fiiiinnnnne.

Much like Gerald in the MWFTW’s example sentence, I found where I could end those fulfilling but unfulfilling things to make room for the new motion in my life, something that would hopefully reach 💯 on the fulfillment scale. And that is precisely when Kate Loves Math* was born.

It definitely wasn’t an easy decision. I very intentionally put the brakes on, for now or maybe forever(?), physically volunteering with causes that are important to me in my community; this is the first time in 15 years that I’m not on a non-profit board. And tho I feel that my work with each of those orgs was impactful, I quite consciously transferred the power I had been giving to that element in my life, that part of my identity(!) and have now put it behind this mathtastic creation that also fills my heart — just in a different way. I made viable space for it and power to it.

We’re headed on into the reflective time of year. We’ve just completed the first week of November, which then means December and the end of 2022 (excuse me, what?) is almost upon us. I know you’re gonna naturally reflect on what is and isn’t important to you and you may consider pushing out the unimportant things in the coming months.

BUT, I’m truly curious to know: Do you feel fulfilled and unfulfilled? Are you fiiiinnnnne? Do you recognize this cyclic way of living? Are you doing something that you’ve given power to, that you would totally define as important…and realize that if you stopped its recurring motion that that would make room for something that’s been in the background in your life?

There’s no right or wrong answer, for sure, I’d just love to know — whatcha gonna do?

October 25, 2022: ADDITIVE

This is how math is everywhere, and in all that we do (all that we ALL do).

As I’m writing this, I am listening to the live Emergency Special Council Meeting of the Portland (Oregon) City Council. If you’re not familiar with what Portland is like right now, there is a heartbreaking amount of houseless folks within the city (and as I just wrote that sentence, I heard that there are currently over 700 encampments within the city’s limits, wtAf?); this emergency meeting is purportedly focused on this issue.

As I listen to all the stats being quoted and the hum of endless politicking as each council member takes the mic, my choice for this week’s Mathnificent Word feels quite appropriate. This new(est) proposal by the city seems to invite a more collaborative (instead of dictatorial) approach for our local government & social service entities.

The words ‘distribution’ and ‘cumulative’ are key to what differentiates how this word ‘additive’ is so much more significant compared to an action that is merely an addition. When you employ an action that is additive, there is intention from the get-go to touch all the entities currently involved plus thoughtful intention to include the entities that need to be involved, and the entire process continues onward together to build, in collaboration, towards one defined goal. Like sprinkling an additive powder over everything, the intention is to give strength and bind all things together so that there is effective on-going communication towards the common end goal, be it making you – or in this case, an entire city – healthier.

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