# Learning Math like Euclid

I'm posting this here as it's off-topic for the Math stack exchange, and I'm hoping there will be some educators here that can point me in the right direction.

I enjoyed watching the YouTube series Numberphile, as it makes math accessible to me in a way it hasn't been before. Which is awesome; I always sucked at math. Anyway, there's an episode where they mention that in Euclid's time, they would learn about shapes and geometry as the basic elements of mathematics and numbers were simply the result of that.

That made me think about the one time I really shined in math classes. Geometry. I was always slow to learn math concepts, as I found the procedural "Just do it like this and you'll get the answer" teaching boring and rote. But not geometry! You can see the relationships and you do things a certain because they make sense. I mean, I'm sure those procedures they were teaching before also made sense in a manner that wasn't yet accessible, but geometry just naturally made sense. Other aspects of mathematics just don't click until I reach a certain point of intuitive understanding (which is why I love Numberphile so much, as it's full of visuals that illustrate relationships between numbers).

Anyway, where can I find resources to learn more in this visual relationship-based manner? I took a pre-calc class in college, but had to drop it when the concepts just weren't clicking. Everything was procedure-based, and I need to find resources that help me to understand the why before teaching the how.

• "Learning Math like Euclid." If you mean that literally, perhaps you can read Euclid's Elements. One of the most influential works in human history. Commented Aug 30, 2017 at 14:38
• You may be disappointed to discover that the main novelty in Euclid's and other ancient Greek books on mathematics is introduction of proofs. Mathematics is about proofs; visuals are aids for reasoning; procedures are practical results proved to be correct. You may want to read answers to math.stackexchange.com/questions/1530840/… . It is about visual learning too. However, you can't learn everything like that. Visuals are just aids. Commented Sep 10, 2017 at 9:50

There are a number of efforts to utilize the power of visualization to convey various aspects of mathematics, often far from geometry. Permit me to make four recommendations, three of which are repost details:

(1: MSE post) Tristan Needham's Visual Complex Analysis, 1999, Oxford Univ. Press.

(2: MESE post) Nathan Carter's Visualizing Group Theory, 2009 (MAA link).

(3: MO post) Bryant and Sangwin's How Round Is Your Circle, 2008:

(4) Martin H. Weissman's An Illustrated Theory of Numbers, 2017, (AMS link):

• I am teaching out of Weissman's book now: it is fantastic. I am doing a reading course with a student on Needham's book (which is also very good, but somehow leaves out phase plots!). I guess I have to check out your other two suggestions now. Commented Sep 7, 2017 at 1:18
• @StevenGubkin: I am envious! I look forward to hearing what you think of Weissman's book further into the semester/quarter. Commented Sep 7, 2017 at 1:31
• @StevenGubkin: This comment is much too late for your reading course, but for phase plots you might find Wegert's Visual Complex Functions useful end enjoyable.
– J W
Commented Sep 10, 2020 at 12:55
• @JW If you like phase plots, check out kisonecat.com/phase-plot, made by my friend Jim Fowler. Commented Sep 10, 2020 at 13:10
• @StevenGubkin thanks for that link!
– J W
Commented Sep 10, 2020 at 14:31

For more geometry, you might enjoy Paul Lockhart's book, Measurement.

For a very common sense approach to a number of math topics, you might enjoy Steven Strogatz's book, The Joy of X.

I have a list of my favorite math books at my blog, and you might find others that are appealing to you.

To me, precalculus has a few main themes. One is understanding how equations and graphs go together, a very visual topic. Another is trigonometry, also a visual topic. For the first, play around with graphing different functions on desmos. For the second, there are lots of cool gifs online that can help you see relationships.

• Side-note, have you read Lockhart's (2017) Arithmetic? It came out ~ 2 weeks ago, but I'm curious about how it compares to e.g. Measurement... Commented Sep 6, 2017 at 20:54
• No! Thanks for the heads up. Commented Sep 7, 2017 at 23:08