Pi is 3. Pi is 4. Pi is infinitely many values because there are infinitely many geometries (1). In the normal Euclidean geometry taught in the core curriculum, we learn that pi is 3.14, but that’s specific to Euclidean geometry.
Euclidean geometry is the geometry of flat surfaces. It makes up our points, lines, flat distances, circles, and squares; basically, anything that can be drawn on a flat surface (2). Any other geometry is a non-Euclidean one. A main axiom, or rule, of Euclidean geometry is that two triangles are congruent if they have matching side-angle-side properties, or SAS (3). In some geometries, the properties of congruent triangles fail SAS, so they cannot be Euclidean. Now that I’ve told you that geometry isn’t exactly the strict constant you learned about in high school, let me explain what pi really is. Pi is just the ratio between the circumference and diameter of a circle, so it’s called the “circle constant.” The circles in Euclidean geometry show that pi equals 3.14, but other geometries have different looking circles, so pi might be different.
An example of a geometry with a different pi is Taxicab Geometry. Taxicab geometry is a geometry with a grid, so think of drawing all your shapes and lines on graph paper (2). The crux is that you cannot go through the squares on the grid diagonally. You have to go along the lines instead of through the squares. This is used for city planning. If you’re traveling in a taxicab, you can’t go diagonally across the city to get somewhere, even though that would be the fastest route. You have to go up and across city blocks and around buildings to get places.
To draw a circle in Euclidean geometry, you simply extend lines from the center of the circle that equal the radius and then connect the outside points. You can’t do this in taxicab geometry, though, because you can’t draw diagonal lines (1). So, if you want to draw a line that isn’t perpendicular to the center of the circle, you have to find the points radius units away from the center and go along the outside of the squares on the grid. If you do this in taxicab geometry, you get a square (2). This means that in taxicab geometry, a circle resembles a Euclidean square. If a circle does not have the same properties as it does in Euclidean geometry, pi cannot equal 3.14 because the circumference and diameter of the circle are different. If you divide the circumference of a circle by the diameter in taxicab geometry, the constant you get is 4 (1). In Taxicab geometry, pi is 4, which means circles are square.
(1) Lewis, Hazel. “Taxicab Geometry.” Maths Careers, http://www.mathscareers.org.uk/article/taxicab-geometry/. Accessed 12 August 2018.
(2) Köller, Jürgen. “Taxicab Geometry.” Mathematische-Basteleien, http://www.mathematische-basteleien.de/taxicabgeometry.htm. Accessed 12 August 2018.
(3) “Elements Book 1.” IIT, n.d., http://mypages.iit.edu/~maslanka/CongruenceCriteria.pd