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This post is inspired by the Quora thread on interesting functions to plot.

**The butterfly**

This is a slightly simpler version of the butterfly curve which is plotted using polar coordinates on Desmos as:

Polar coordinates are an alternative way of plotting functions – and are explored a little in HL Maths when looking at complex numbers. The theta value specifies an angle of rotation measured anti-clockwise from the x axis, and the r value specifies the distance from the origin. So for example the polar coordinates (90 degrees, 1) would specify a point 90 degrees ant clockwise from the x axis and a distance 1 from the origin (i.e the point (0,1) in our usual Cartesian plane).

2. **Fermat’s Spiral**

This is plotted by the polar equation:

The next 3 were all created by my students.

3. **Chaotic spiral (by Laura Y9)**

I like how this graph grows ever more tangled as it coils in on itself. This was created by the polar equation:

4. **The flower (by Felix Y9)**

Some nice rotational symmetries on this one. Plotted by:

5. **The heart (by Tiffany Y9)**

Simple but effective! This was plotted using the usual x,y coordinates:

You can also explore how to draw the Superman and Batman logos using Wolfram Alpha here.

**Euler’s 9 Point Circle**

This is a nice introduction to some of the beautiful constructions of geometry. This branch of mathematics goes in and out of favour – back in the days of Euclid, constructions using lines and circles were a cornerstone of mathematical proof, interest was later revived in the 1800s through Poncelot’s projective geometry – later leading to the new field of non Euclidean geometry. It’s once again somewhat out of fashion – but more accessible than ever due to programs like Geogebra (on which the below diagrams were plotted). The 9 point circle (or at least the 6 point circle was discovered by the German Karl Wilhelm von Feuerbach in the 1820s. Unfortunately for Feuerbach it’s often instead called the Euler Circle – after one of the greatest mathematicians of all time, Leonhard Euler.

So, how do you draw Euler’s 9 Point Circle? It’s a bit involved, so don’t give up!

Step 1: Draw a triangle:

Step 2: Draw the perpendicular bisectors of the 3 sides, and mark the point where they all intersect (D).

Step 3: Draw the circle through the point D.

Step 4: From each line of the triangle, draw the perpendicular line through its third angle. For example, for the line AC, draw the perpendicular line that goes through both AC and angle B. (The altitudes of the triangle). Join up the 3 altitudes which will meet at a point (E).

Step 5: Join up the mid points of each side of the triangle with the remaining angle. For example, find the mid point of AC and join this point with angle B. (The median lines of the triangle). Label the point where the 3 lines meet as F.

Step 6: Remove all the construction lines. You can now see we have 3 points in a line. D is the centre of the circle through the points ABC, E is where the altitudes of the triangle meet (the orthoocentre of ABC) and F is where the median lines meet (the centroid of ABC).

Step 7: Join up the 3 points – they are collinear (on the same line).

Step 8: Enlarge the circle through points A B C by a scale factor of -1/2 centered on point F.

Step 9: We now have the 9 point circle. Look at the points where the inner circle intersects the triangle ABC. You can see that the points M N O show the points where the feet of the altitudes (from step 4) meet the triangle.

The points P Q R show the points where the perpendicular bisectors of the lines start (i.e the midpoints of the lines AB, AC, BC)

We also have the points S T U on the circle which show the midpoints of the lines between E and the vertices A, B, C.

Step 10: We can drag the vertices of the triangle and the above relationships will still hold.

In the second case we have both E and D outside the triangle.

In the third case we have E and F at the same point.

In the fourth case we have D and E on opposite sides of the triangle.

So there we go – who says maths isn’t beautiful?