# Topic of presentation: Elementary Geometry vs Fibonacci & its sequences

#### mathmari

##### Well-known member
MHB Site Helper
Hey!!

Between the following two topics:
1. Elementary Geometry
2. Fibonacci and its sequences
which would you suggest for a presentation? Could you give me also some ideas what could we the structure of each topic?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hey mathmari !!

Sun flowers!
And various other parts of nature.
They have Fibonacci's numbers embedded in them, and the ratio approaches the golden number, which is also a nice exercise in elemental geometry where we can also see the golden number.

#### mathmari

##### Well-known member
MHB Site Helper
Which of them do you think is more interesting and better for a presentation?

For the Fibonacci numbers we could refer the sequence and the formula, some applications, some properties, or not?
For the elementary geometry we could refer to the properties of straight lines, circles, planes, polyhedrons, the sphere, the cylinder, or not?

Do you have an other better idea?

#### topsquark

##### Well-known member
MHB Math Helper
Just to give a though for the other possibility. With elementary geometry you can discuss the Euclid's axioms and postulates. The parallel postulate is always good fun as so many have tried to prove that it doesn't need to be included. (It does need to be because it's a launching point for non-Euclidean geometries.)

Lots of fun stuff you can talk about.

-Dan

#### mathmari

##### Well-known member
MHB Site Helper
I looked for both topics and I think Fibonacci is more specific, elementary geometry is a more abstract topic, isn't it?

As for the Fibonacci one, what do you think about the following structure:

1. An introduction about the topic
2. A little biography of Leonardo Fibonacci
3. Some words about the Fibonacci sequence
4. Some properties about the Fibonacci sequence
5. Applications

#### Klaas van Aarsen

##### MHB Seeker
Staff member
What would you put in section 3?
Which applications in section 5 are you thinking of?

Btw, if it were me, I'd include a couple of neat videos.
For starters one in the introduction - to immediately grab the attention of the audience.
And more videos in other parts of the presentation.
There are some very nice videos around that show how Fibonacci appears in nature.
I'd also highlight the connection to the Golden Ratio, which ties it to elementary geometry as well.
That may deserve its own section.

#### mathmari

##### Well-known member
MHB Site Helper
Ok! As for the begining, the structure could be the followinf, or not?

- Definition/Formula
- Using other inital values we get the lucas sequence
- Example of application of Fibonaccisequence
- Properties of Fibonacci/Lucas-sequence and proving some of them of mention just the idea of proof
- Formula of Binet to get the explicit definition
- Relation between fibonacci and lucas sequences
- The sequencews are also defined for negative indices

What do you think of that? Or could we d that better?

The given notes for the properties (this is the first part) are here (they are in german).

Then as for the second chapter, it is about Fibonacci and Linear Algebra, here are some notes.
Here again we coud mention all the properties and for some give also the proof.

What do you think of that?

Last edited:

#### Fantini

MHB Math Helper
Mathmari, what is the target audience for this presentation? I don't think I've seen you mention this. It matters a lot when it comes to structuring your talk.

#### mathmari

##### Well-known member
MHB Site Helper
Mathmari, what is the target audience for this presentation? I don't think I've seen you mention this. It matters a lot when it comes to structuring your talk.
The audience consists of the other students of that lecture.

#### mathmari

##### Well-known member
MHB Site Helper
I thought now about elementary geometry, and more precisely on the straight line and the triangle.

For this I use the book "Elementary Geometry" by Ilka Agricola, Thomas Friedrich (chapters 1.1 and 1.2).

The following topics are discussed there with regard to the straight line:
- Intercept theorem
- Pappus's hexagon theorem
- Desargues's theorem
- Theorem of Thales

And regarding the triangle:
- Theorem: A triangle is isosceles if and only if two of its inner angles are equal.
- Theorem: A triangle is equilateral if and only if its three interior angles are equal.
- Exterior angle theorem
- Sum of angles in a triangle
- Alternate angle theorem

So would the structure of the presentnation mention all of these topics and prove some of them? Or what do you think?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Ok! As for the begining, the structure could be the following, or not?

- Definition/Formula
- Using other inital values we get the lucas sequence
- Example of application of Fibonaccisequence
- Properties of Fibonacci/Lucas-sequence and proving some of them of mention just the idea of proof
- Formula of Binet to get the explicit definition
- Relation between fibonacci and lucas sequences
- The sequences are also defined for negative indices

What do you think of that? Or could we d that better?

The given notes for the properties (this is the first part) are here (they are in german).

Then as for the second chapter, it is about Fibonacci and Linear Algebra, here are some notes.
Here again we coud mention all the properties and for some give also the proof.

What do you think of that?
The audience consists of the other students of that lecture.
I think it is a lot to cover in a single presentation.
To be honest, I'm not really familiar with the Lucas sequence, and there is quite some information there that is not known to me yet.
Still, that might actually make it interesting to an audience that is already familiar with Fibonacci in general.
It does seem to me that it is too much.

Since your structure is basically the first chapter from a book, it seems to me it would effectively be a lecture in a teaching course.
Is that what you intend?

I thought now about elementary geometry, and more precisely on the straight line and the triangle.

For this I use the book "Elementary Geometry" by Ilka Agricola, Thomas Friedrich (chapters 1.1 and 1.2).

The following topics are discussed there with regard to the straight line:
- Intercept theorem
- Pappus's hexagon theorem
- Desargues's theorem
- Theorem of Thales

And regarding the triangle:
- Theorem: A triangle is isosceles if and only if two of its inner angles are equal.
- Theorem: A triangle is equilateral if and only if its three interior angles are equal.
- Exterior angle theorem
- Sum of angles in a triangle
- Alternate angle theorem

So would the structure of the presentnation mention all of these topics and prove some of them? Or what do you think?
Again, it may be too much.
A presentation that covers all of it, may be rattling through the material.
Then it would only be understandable to an audience that already knows all of it.

What is the purpose of the presentation?
If the target audience are other students of the same lecture, then that sounds as if it is a teaching exercise.
Is it?