euclidean geometry

Euclid’s first postulate

postulate1

To draw a straight line from any point to any point.

A postulate, to quote from dictionary.com is something taken as self-evident or assumed without proof as a basis for reasoning. And according to Euclid, nothing can be more self-evident than the fact that given two points, one can draw a straight line segment joining them.

Now, it is also obviously self-evident that there is exactly one such line joining two given points, i.e. that there is a unique line joining any two points.

However Euclid does not mention the uniqueness, though he seems to assume it in the subsequent proofs .. So probably a more accurate first postulate would be

To draw a straight line (which will be unique) from any point to any point.

The Processing code

int time1 = 500;
   int time2 = 2500;
   PFont f;
   
   void setup() {
     size(600, 120);
     smooth();
     f = createFont("Arial",20,true); 
}
   void draw() {
     int currentTime = millis();
     background(204);
     if (currentTime > time2) {
       ellipse(100,60,10,10);
text("A",95,90);
  fill(0);
ellipse(500,60,10,10);
text("B",495,90);
       line(100,60,500,60);
     } else if (currentTime > time1) {
       textFont(f,20); 
         fill(0);
ellipse(100,60,10,10);
text("A",95,90);
  fill(0);
ellipse(500,60,10,10);
text("B",495,90);}
    
   }
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A series of posts on Euclid’s Elements

Why, pourquoi ?!

OK, this series of posts on Euclid’s Elements serve a dual purpose.

  1. To recall my vague memories of Euclidean geometry, the subject that first made me like mathematics in general
  2. To learn animation using the software called Processing. So the terrible gifs that you see here record my baby attempts at learning to draw geometrical figures with some time lapse on the computer. I’ve decided to go the immersion route, … meaning I have a clear cut goal of exactly what gif I want to make and I will make it even if it means I write the dumbest code to get it working. Hopefully, as time progresses, I will get better at figuring out more efficient and elegant ways to do it.

Is each post going to be bit-sized or byte-sized or giga-byte sized ?

I hope to be very brief with the text and let the gifs do the talking. You know, “Show, don’t tell”. Also, the posts will mostly be distraction free, so you won’t have a link every second sentence.

I vaguely know what an isosceles triangle is and that parallel lines never meet. Can I still follow what’s going on or are you going to talk Latin ?

I guess ‘talking Greek’ would be a more appropriate phrase since Euclid wrote his Elements in Greek. But leaving frivolity aside for a second, the beauty of Euclid’s elements is that he builds everything from scratch. So you don’t need to know anything to begin with. However, each step will heavily rely on previous steps, so it might be slightly advantageous to go in sequential order

Who are you ? Do you really know Euclidean geometry ? Can I trust what you are saying ?

Well, I’m a working mathematician. I did know some Euclidean geometry when i was in high school. Sadly due to lack of use, I have forgotten many of the cool things I used to know, so hopefully this will be a me-and-you-learning-together.

You should never trust what is being said blindly. But I will try prove most of the things I say here, so trust me after reading the proofs!

What is Processing ? 

It is a software where you write some code to draw/animate some nice figures on your computer. I’m a novice at programming and helpless with technology, though I can write some if and for loops in various languages. So don’t ask me more.

 

Will you post your code for the gifs that you make ?

Yup

Are the posts going to be periodical ? 

No. It ought to be, I wish it could be, but I’ll probably end up doing this once in a blue moon when I have some time to kill..

Are criticisms welcome ?

Yes. Also error correction is doubly welcome

Is this pointless post ever going to end ?

Point taken. Until the next time!