Spirals can be seen in many natural phenomena (snail shells, flower petals, etc.). One specific spiral is the Vogel Spiral defined by the following polar equations:

𝜃 = i * 137.507°

r = sqrt(i / n)

Where n is the number of points to be generated and i is the index of a certain point in the spiral and the angle 137.508° is the golden angle. Different angles produce different patterns of florets of the spiral.

Since we're dealing with polar equations I'm going to start off with this blog post, and make few changes to produce the Vogel spirals.

The setup was as per the below screenshot.

By changing the number of points to be generated and the angle we can create different beautiful Vogel Spirals such as the ones below:

If you've been following my work you may have noticed that I love math--for its beauty and elegance. But I love math more when I can find practical applications in my work.

Here is a real case. Sometimes I need to generate dots or icons to represent a crowd, a sample or a species--say for example a sample of 100 employees. Instead of drawing one shape--say a dot--and then copy/pasting it 99 times and trying to randomly positioning those dots on the slide, I can use the above template to generate such graphics as below. The dots look random and more natural.

Or I can produce a crowd by inserting an icon onto the sheet, resizing it and then copy/pasting it into the chart as follows:

For producing the Vogel spiral I don't need the lower and upper bounds. I just need to produce the i's and have the initial angle (137.507°) as an input.

For that I'll do the following:

1. Insert a column at the start of the sheet.
2. Enter "i" in cell A7 and "Angle" in cell C5.
3. Delete the content of the cells in the range C2:D3.
4. Enter the number 256 in cell D4.
5. Enter the following formulas in the respective cells:

D5 -- > = 137.507 * PI() / 180
A8 --> = SEQUENCE(D4,1,0,1)
B8 --> = D5 * A8#
C8 --> = SQRT(A8# / D4)

6. Change the chart type to Scatter and reset the boundaries of the vertical scale.
7. Delete the chart title and the worksheet should look like this:

Views > Excel charts > Maths in excel the vogel spiral phenomenon

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Here is another one with 195 foxes:

Thank you for reading, and I hope you found this useful. If you have any questions, find me on X and ask me anything.