Tag Archives: math

Map of Mathematics

I came across this Map of Mathematics in a TikTok where people working in the field of Mathematics were all sharing how their fields intersected different parts of the map. Here is the video:

I love how the concepts and fields of study connect and interconnect.

The the video starts with a ‘stitch’ of a Tiktok user asking, “Did everyone know that Mathematicians are, like, discovering new Math?”

This plays into a recent post in which I state that Math is ‘Discovered more than invented‘. There is so much more to discover, and I believe, thanks to my conversations with Joe Truss, that geometry is the foundation of Math. But math is a hard conceptual language with which to understand the geometry… and if we study the geometry we can understand the universe without having to do the hard math. Oh, and if Mathematicians better understood the geometry they could discover new Math a little faster.

In the Map of Mathematics, Geometry is the key.

Discovered more than invented

Is Math invented or is it discovered? Is Math a human, or at least biological construct that helps mammals understand the world we live in? Or does it exist inherently in the universe and is it revealed to us through curiosity and scientific discovery?

I’ve shared my fascination in playing, learning, and discovering secrets held in Geometry (here and here) with Joe Truss.

Recently I watched a video that quoted Galileo Galilei:

“Mathematics is the language with which God has written the universe.”

With everything Joe and I have been playing with, I thought of an adaptation of this quote:

The universe is written in the language of geometry, and Mathematics allows us to measure it.

When I shared this with Joe, he tweaked it:

The universe is written in the language of geometry, and Mathematics allows us to take its measure.

I like this because it speaks to the idea of Math being discovered, of it being inherent because it is present in the geometry. It is Geometry that is the underlying language. Math is the alphabet, the universe is written in geometry. This is why Pi pops up where you don’t expect it; why the Fibonacci sequence is prevalent in nature; why we can land a ship on the moon and Mars; and even why our religious symbols come out of geometry.

The challenge we have is that we don’t fully understand the geometry, sometimes rather than discovering the true essence of the mathematics, we have to invent approximations that help us make sense of universe. Understand the geometry and maybe we can remove infinities and irrational numbers in our calculations… Understand the geometry, and the math becomes more accurate. Understand the geometry and maybe we can unify our theories that currently seem to contradict themselves.

Discover the geometry and we discover the math behind it, no longer needing to invent approximations of the math to translate the geometry into a world of calculations and numbers that don’t quite fit with they universe we are trying to describe. When we discover the geometry, we no longer needs to invent Math, the geometry allows us to take its measure… And in fact we can explain the universe with the geometry, and then we don’t have to be a physicist or mathematician to understand it.

Playing with geometry

The image below is a blue cube octahedron with each triangular face forming a tetra-octa-tetra pattern coming out from the center.

This is a tetra-octa-tetra, with two yellow tetrahedrons on opposite sides of an octahedron:

I’m fascinated by these shapes thanks to my uncle, Joseph Truss, who has been playing with these shapes for decades, and who believes the fundamental building blocks of our universe is the tetrahedron. We live in a ‘Tetraverse’.

But the geometry he describes goes far beyond this. He intuitively understands the underlying structure of the universe through geometry in a way that mathematical physicists understand it through math.

Watch this 28 minute movie, Hacking Reality. It gets to some of the topics Joe and I discuss. I really think he’s on to something some of the smartest physicists in the world are still trying to figure out… but he’s done it purely by playing with geometry, and visually/intuitively making sense of how shapes are formed and come together.

Fun with Math

Take ANY number, subtract the digits in that number, and the answer is divisible by 9.

14… 14 – 1 – 4 = 9

25… 25 – 2 – 5 = 18

36… 36 – 3 – 6 = 27

49… 49 – 4 – 9 = 36

1,234… 1,234 – 1 – 2 – 3 – 4 = 1,224

And you don’t need your calculator to figure out if 1224 is divisible by 9, it is. How do I know? Add 1 + 2 + 2 + 4 together and if it’s 9 or divisible by 9, then the number is also divisible by 9.

7,692… 7,692 – 7 – 6 – 9 – 2 = 7,668

And then 7 + 6 + 6 + 8 = 27 which is divisible by 9 so 7,668 is divisible by 9 (= 852).

I love Math.

The answer doesn’t seem right

Back in December I shared that we really don’t understand exponential growth. Well I’m about to share a question with an answer that will boggle your mind. All I ask is that you make a guess at the answer before looking at it.

Imagine we get every single human being on earth to play a ‘winner play on’ game of rock-paper-scissor. First, let’s round up the population to 8 billion. And let’s assume every one of them can play.

A game of Rock-Paper-Scissors with 8,000,000,000 people.

So the first round will be 4 billion pairs of people, with 4 billion losing, and 4 billion winning and moving on to the next round. The question is:

How many rounds of rock-paper-scissors will the winning person have to win consecutively in order to be the world champion?

Guess.

I’ll give you a hint: it’s less than 100.

What do you think?

Here is the answer.

Well, what if we added another 1/2 billion people? That would be 500 million more players, or 8.5 billion total. The answer would still be the same!

Hard to believe. It’s so difficult to wrap your mind around needing this many games to whittle 8 billion people down to one winner. It just doesn’t seem right… but it is.

The real numbers

In the last week and a half I’m aware of 18 people whom I personally know have caught covid-19, likely the Omicron variant. But of these 18, (half of which are 2 families), I think only one of them is reported. Others sat on 811 for more than 2 hours before giving up on the call. Others have mild symptoms, and here in BC they are asking people with mild symptoms not to get tested, because testing locations are being overwhelmed.

Extrapolating on these numbers, even if I overestimated the amount of cases in BC to be 1/3 reported (that would be 6/18 rather than 1/18), then the real number of new covid cases is a massive number.

This is surprisingly a good thing. First of all, our hospitals are not overwhelmed with serious cases. Secondly, at this rate of infection, we are moving quickly to heard immunity.

Well, that’s my hope anyway. I am only working with my own anecdotal evidence, and coronavirus has kicked my predictions in the butt more than once. Still, it’s good to look forward and see a little light at the end of the tunnel.

Maybe by spring everything will be open and masks will be optional. Maybe. Still, it feels good to have a little optimism right now. Even if we really don’t know the real numbers, I think they are looking good… Time will tell.

Doing STEM

‘Doing STEM’ or ‘Doing STEAM’… there is a saying, “Put lipstick on a pig, and it’s still a pig.”

I don’t want this to sound like a rant, and I don’t want to knock teachers for trying to do STEM projects. I do want to say that if 5 years ago a teacher did a project with kids where they broke them into groups and had them assemble a limited number of straws and a specific length of tape into the tallest possible tower, and if they do it again today it isn’t suddenly a STEM project.

Now, if that same lesson included teaching geometry and/or structural integrity; or if students had to design it such that their design had to have a function such as offices or apartments; or hold a weighted satellite dish; or if it had to factor in wind resistance (such as a blow dryer at close range); or if they had to model their design first and estimate the height they will achieve… if there was some thinking, designing, modelling, or estimating that was required before or even during the build process, well then it’s looking more like STEM.

Hands-on does not equal STEM. Building something does not equal STEM. Group challenges does not equal STEM. Meaningful integration of cross-curricular concepts, where problem solving requires thinking in more than one subject area relating to Science, Technology, Engineering, and Math is STEM. It doesn’t have to check all the boxes, but it should include thoughtful integration of at least a couple of these.

It’s about making the cross-curricular connections explicit, or at least thinking through how the outcomes and expectations relate to these connections. It’s about developing competencies in the areas of STEM and not just doing a project that looks like STEM.

We don’t really understand

We don’t really understand exponential growth. It’s too hard to comprehend because when we look at growth, we tend to focus on what we’ve seen already, and project forward, but what has already happened is always less significant in length or size than what is still to come. So when we compare what has happened already to what is still to come, we are not comparing equal things.

Fold a piece of paper in half 6 times. How thick do you think the stack would be? Let’s have some fun and look at the folding paper challenge:

It was an accepted belief that folding a piece of paper in half more than 8 times was impossible. On 27 January 2002, high school student, Britney Gallivan, of Pomona, California, USA, folded a single piece of paper in half 12 times and was the first person to fold a single piece paper in half 9, 10, 11, and 12 times. The tissue paper used was 4,000 ft (1,219 m; 0.75 miles) long. ~ GuinnessWorldRecords.com

So she needed a 4,000 foot, (1,219 metres) long piece of paper to achieve this. It’s easy to look at this image of her folded paper and figure out how big it was at 11 folds and before that 10 folds, by halving the amount once then twice. But what if she were to fold the paper more times? How many more times would this image represent?

This image represents folding the paper just 3 more times… a total of just 15 folds.

At 23 folds this would be about a kilometre high (3,280 feet). At 30 folds, you would be entering space. 42 folds gets you to the moon. The 51st fold would get you to the sun. Beyond that it doesn’t matter because our brains won’t truly appreciate the scale anyway.

So I can see the difference that folding a piece of paper just 6 times (64 pieces of paper high) to 12 times (the first image of Brittany above) looks like, but I really struggle to extrapolate from this that 24 folds would be 2 kilometres high.

So when we look at things like technological advancements, we don’t really see well into the future. When I bought the 16k adapter for my Commodore VIC 20 computer to get me to a whopping 36k of memory, I could not fathom the idea that I’d one day be buying 2 Terabytes of memory to store photos that were 8 megabytes large. And I’ll have an even harder time imagining what kind of data I’ll be storing 10 or 20 years from now.

Watch out Metaverse here we come! What does this mean? It means that in 20 years we’ll look back at the technology we have right now in the same way someone who lived 160 years ago would look at our technology today.

That’s mind blowing!

By the numbers

Have a look at this short visual presentation:

Putting COVID-19 Numbers and Vaccinations into Context

It is an insightful, graphic way to look at what has happened to death rates of covid patients since the vaccine was introduced. It helps you understand the scale of this compared to things like yearly deaths by lightning, drowning, and traffic accidents. But these two images really put things into perspective:

Have a look at Covid deaths in the US, especially after vaccine introduction (the box on the right):

And how things would look if vaccines didn’t work, versus the actual statistics:

It’s worth going to the full, short, easy to consume slide show, and grasping all the information shared, by the numbers.

By the numbers

It’s staggering to look at the hospitalization numbers coming in, now that millions of people have been vaccinated. It’s simple: get vaccinated and you are extremely unlikely to end up in the hospital even if you still catch the Delta variant. But what are the motivations of the unvaccinated to change their minds? Travel. Not safety, not protecting others in the community, but the fear of travel restrictions without proof of vaccination.

I don’t watch TV but I’ve been going on TikTok for a 1/2 hour or less, (I keep a time limit in this addictive app), and I follow a lot of doctors and epidemiologists on the site. What I find is that about 1/2 of their posts are about research results and the other half tend to be responsive to comments they get on previous posts. Some of these questions are good, even if they question the results. Some are responses to asinine ignorance.

Many people don’t know how to do, or understand, the necessary math. Here is a fictitious example of a deceiving headline: “50% of new cases are vaccinated“. That sounds like the vaccine isn’t working. But later in the article it says that 80% of the population is vaccinated. Making the math simple, if there were 100 people and 4 got covid, that would mean two non-vaccinated and two vaccinated people contracted the virus (50% each). But the population was 20 unvaccinated and 80 vaccinated, so while the cases were 50-50, the chances of getting covid work out like this:

Unvaccinated: 2/20 = 10%

Vaccinated: 2/80 = 2.5%

While the headline makes it seem like the vaccine isn’t working, the unvaccinated are four times more likely to contract the virus in this example. This doesn’t even factor in the significant increase of risk of hospitalization/death for the non-vaccinated.

Right now two things are causing headlines like this to proliferate:

1. A balance of ignorance and desire to get clicks and views: Reporters not understanding the math themselves and rushing to get the attention-gaining headline out.

2. Deliberate malice: People who know exactly what they are doing and want to promote an anti-vax narrative.

As we move forward and the numbers will start to be more available as well as more obvious. Will that make a difference? Probably not for those that are entrenched. It’s not the numbers that will change people’s minds, it will be the (proclaimed) limitations on their liberties, like travel… and buckle up because this isn’t going to happen without a lot of whining and complaints.

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Related: I’d rather be a sheep than a lemming