**Why Horizontal?**

*Blog by Ioanna Georgiou and illustrated by Asuka Young*

The section in red comes from Season 9, Episode 15 of the TV series “Friends” created by Marta Kauffman & David Crane, and broadcasted by NBC between 1994 and 2004 and readily available on streaming channels. We use the everyday mathematical themes mentioned as our starting point, to explore the fascinating world of mathematics further.

Joey: Hey

Monica: Hey, how did the audition go?

Joey: They want to see me again this afternoon but, boy, Leonard Hayes did not like me!

Monica: What happened?

Joey: He said I was not urgent enough; you know and that everything I did was horizontal (gesturing vertically) and I should be more vertical (gesturing horizontally), oh and he said I should think less.

Monica: So far so good.

Thinking time: Who invented the axes and why? What are we using them for? What could Leonard Hayes mean when he said that the acting should be less horizontal and more vertical?

It is quite amazing to think that all the properties we experience with shapes (right angles, lengths, parallel lines, and so on) can be translated into some sort of algebraic language as well.

This is relatively old news of course, one of the pioneers of this approach was the “I think therefore I am” French philosopher and mathematician René Descartes (1596-1650). Whilst there might be some dispute on whether “I am” requires “thinking” (can’t we just be?), Descartes is credited with the creation of the horizontal and vertical axes, or as we know and love them, the *x*- and *y*- axis. The plane is called the *Cartesian plane* after Descartes. Whoever named it, did not like the *Des *bit. Clearly.

To get to any point on a surface, we need two pieces of information. We know this from experience. When Alexandra got a call from Sanshiro to meet up and only told her the number the café was at, she was well lost, impossible to find the place. On another day, just for fun, Alexandra gave Sanshiro only the street name of the café they were meeting. It would need to be an extremely short road for Sanshiro to find the (possibly only) café. He would need both, street name, and number. They eventually figured it out by dropping a pin on google maps, which incidentally uses two coordinates, latitude, *and* longitude.

Similarly, to find a point on the *x-y* plane, we need two coordinates, *x* and *y*. We write them as a pair in brackets, with x always first. It looks like this: (*x, y*) . It is a happy coincidence that the order in which we write them is also alphabetical (x comes before y in the alphabet) so no risk of getting confused here!

We can then describe lines using algebraic equations right on this *xy* plane. For example, a horizontal line like the one here clearly shows that the green points, all on this horizontal line, have the same *y*-coordinate. Their *x*-coordinate can vary to pretty much anything, but their *y*-coordinate stays the same. This helps us create a simple equation to describe a line! Awesome! The equations will always be where the number is wherever the line crosses the *y*-axis.

Similarly, we can get any vertical line written as as all the points on the line will have the same *x*-coordinate, but the *y*-coordinate can again vary and take any real value.

Ok what do people mean when they say “real value” or “real number”? The easiest way to remember this is: any number you can find on the number line is a real number. So pretty much any number we encounter in everyday life would be a real number.

The word horizontal comes from the “horizon” which is the imaginary line separating the earth from the sky, and it is, well, a horizontal line! Vertical comes from the Latin verticalis, and it was used to mean directly above, or overhead.

Confusing the meaning of horizontal with vertical was unexpected perhaps due to the word coming from “horizon”. So, Monica’s response about Joey being able to “think” even “less” is hilarious.

But how about what the director meant about the performance and using these rather mathematical words for it? We can take them to mean that a horizontal performance is rather flat (fact) and a vertical performance would be more lively, maybe even intense!

Next steps: See how often these words come into everyday speech. Do you only use them for maths, or do they just go unnoticed? Sketch some horizontals (*x = number*) and some vertical lines (*y = number*) until you are comfortable with these equations!

#### More about the Author and Illustrator - and Preorder the Book

Asuka Young, the amazing illustrator I collaborated with for Mathematical Adventures! and Peculiar Deaths of Famous Mathematicians has livened up the dialogues with her illustrations! And we can’t wait to see this material transforming into a book in the end of this series. You can even pre-order it at a special price if you’d like! See here for more details.