# Why we have to be so precise in Geometry?

Previously I've explained some basic things of graphs to my kid, such as planar, $$V-E+F=2$$.

Now when I introduce geometry, he asked, "Why we have to be so precise in Geometry?"

Indeed, in graphs, vertices are just stones, and edges rubber bands; but things are different in Geometry, edges can't bend, vertices can't just shift.

Why so? How could I explain the constraint?

• How old is your kid?
– user507
Sep 17 '20 at 16:28
• As back to him: "Why do we have to be so precise in cooking times, but not in sleeping times?" Sep 17 '20 at 17:19
• Reminds me of mathematical fallacies in geometry due to imprecise diagram :) Sep 18 '20 at 6:25
• In case anyone was wondering, "As back to him" is not some kind of strange local dialect. Rather, I had intended to write "Ask back to him". (Even this sounds a bit strange, but I was in a hurry.) Sep 18 '20 at 10:49
• Indeed, in graphs, vertices are just stones, and edges rubber bands; but things are different in Geometry, edges can't bend, vertices can't just shift This is a very cryptic statement. It's hard to even understand what mental model you are trying to elucidate here. Could you explain a bit more about what you're trying to say here and how your question relates to it?
– J...
Sep 18 '20 at 16:36

Because the aim of geometry is to study properties related to shape, size and length. Therefore, in the context of geometry, we cannot deform our objects because deformation changes these properties.

On the other hand, the properties in which graph theory is interested are independent of size, shape and length. Therefore, we do not care about them.

(However, it is important to note that, from the mathematical point of view, we are "precise" in both.)

• and this is why during school I often confuse the question when it asked like "given this triangle with a line dividing it into 2 part in the middle calculate the angle", stupid me was just like "well this looks like a 90 degree angle so ...." and I got the wrong answer. Sep 18 '20 at 10:09
• The OP has clarified in comments that their son is 7. To me this answer seems to abstract for a kid that age.
– user507
Sep 18 '20 at 19:27
• @BenCrowell Given the nature of the question, it doesn't seem very abstract, if properly explained: Take a cube and deform it into a rectangular cuboid with half of the height. The kid can see that the number of vertices, faces and edges are the same. Here we have examples of properties that are independent of size, shape and length. The kid can also see that distance between vertices, volume and area are different. Here we have examples of properties that depend on size, shape and length. We have to be "precise" in order to ensure that we will not change the properties we are interested in. Sep 18 '20 at 20:12
• It also goes the other way around: in geometry we do not care about color, in graph theory we (sometimes) do. Sep 20 '20 at 9:33

Geometry was invented because we needed to measure land. If you have a field and want to calculate how many cows can you fit in it (area), you will want to be as precise as possible.

• This is a good answer for a kid the age of the OP's son, who is 7. It's simple and concrete.
– user507
Sep 18 '20 at 19:26
• @BenCrowell How this relates with the fact that we "don't have to be precise" with graphs? The kid is comparing graphs with geometry. Why we don't have to be "as precise as possible" with graphs? Sep 18 '20 at 20:21
• Gets messy if you allow fractional cows. Or circular fields, which result in Cow Pi. Sep 18 '20 at 22:58
• ... but not to (claim to) be more precise than possible. I once saw a map of several neighbouring agricultural villages that gave a value of the area of each village to seven significant figures. Sep 20 '20 at 10:20

Here's an example which shows what happens when we get sloppy with geometry: The missing square triangle.

You can cut the pieces with your kid, and let them try to spot the problem. It might be hard to notice visually, so that's why we use precise fractions or angles.

It's similar to the "Infinite Chocolate Bar Trick", but be careful not to disappoint your kid when the harsh truth is revealed!

Mathematics is about precision and abstraction. You abstract away (i.e., completely ignore) some aspects of your problem to simplify reasoning precisely about those that do interest you.

Geometry (Euclid style, here) is about line lengths and angles and also areas. You disregard line or area widths, colors, textures; consider (mostly) straight lines and circles and their intersections

Graph theory is about objects (vertices) and their connections (edges). You omit everything else.

because when we say 'Geometry' in normal school terms we are referring to Geometry as studied by Euclid's book from ancient Greece, which was based on 5 axioms. it is also sometimes called 'straight edge and compass' geometry. And it was precise by it's design.

The trick is with Euclid's geometry that you are not just studying shapes and lines, you are studying how logic and reason works. It shows how far you can go and what you can accomplish by starting with very good basic assumptions about something, and working out the conclusions based on logic and reason. Philosophy and Geometry used to be very closely related subjects in schools of ancient times.

That is why we still study it even though non-Euclidean geometry has existed for 100 years and numerical approximation is much more important in the modern computerized world. In real life, engineers use computers to calculate tolerances within a precision, objects are never perfectly shaped. And in physics, Einstein's Field Equations are the most accurate known representation of the motion of bodies in the cosmos, and they are not precise in the sense that everything has to be approximated by calculation. There are no orbits that are perfect circles or ellipses, everything is approximated numerically in a computer. But we still study Euclid, kind of like why we still learn the alphabet, study Chess or Go, or still play cards or sports.

When I was at school, both the teachers and the textbooks carefully avoided using the names "edge" and "vertex" in topology, using the names "arc" and "node" instead. That way, no-one got confused by expecting the edges and vertices of 3D geometry to have the same properties as the topological objects.