# Numerate Place

A numerate place refers to a single point. An exact point. It is a reference for where one is or where something is. But how do we define where a point is in space? Space is vast and seems limitless to us in our day to day experience, and a point, being one dimensional, is small, almost to the point of being invisible!

As soon as this point begins to move, it begins to enter into the second dimension. However, it’s place becomes a path, a record of movement through space. Lines are just that, records of a movement. Think of your pencil when drawing. You put it down, create a mark, and then from that mark a trail emerges recording where your hand has gone. Mathematically, we express this y=f(x).

This is a cartesian plane, where you can plot the results of solutions generated by a mathematical expression (i.e. y=f(x)), by identifying the x and y coordinates.

To define a point we need a way of recording where it is in space. Consider what happens when two lines intersect! Two intersecting lines identify where a place is on a planar surface. In mathematics, this is known as the Cartesian plane with the X & Y axis. In a practical application of this concept, we have longitude (east-west) and latitude (north-south) to identify our location on planet earth. Using mathematical notation, we can say that one path is represented by y=f(x) and for another y=g(x). To find the point of interesection we create the expression f(x) = g(x).

Two intersecting parabolic functions.

But paper or canvas does not adequately represent our human experience. We live in a world in which we exist in and move through three dimensional space. X and Y axis do not represent our relatively new found ability to fly – we now have longitude (east-west), latitude (north-south) and altitude (height from the surface from the earth).

The Cartesian Plane in three dimensions utilizing the x axis, y axis, and z axis.

The following video is from Khan Acadamy and explains Rene Descarte’s discovery of the relationship between algebra and geometry, allowing us to visualize complex mathematical expressions and to see patterns in new ways. Or in other words, providing us with a new way of seeing a place within space!

But how do we mathematically solve for a single point in three dimensions (along the x, y and z-axis’)? Unlike solving for a point of intersection using 2 lines, where we can set one equation equal to an other and attempt solving for the variables, in three dimensions you need to solve for 2 of the three variables and check to see if they satisfy the third equation. The following video helps to explain how this is done!

### Guiding Questions

1. Identify additional mathematical concepts that relate directly or indirectly to the artistic theme of place.
2. Explain how mathematicians use skills developed in art to help illustrate their thinking.
3. Explain how artists may use mathematical concepts when developing their artworks!