Excerpt from the book Philosophy for Heroes
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What does math have to do with becoming a hero?

We live in a complex world, but it is possible to understand even “irrational” aspects of it.


What is the connection between mathematics and heroism? At first, the two hardly seem related--math is about measuring things, and heroes do immeasurable good. Isn’t one limited and the other limitless?

But the two are, in fact, connected. To become a leader, which is the foundation of being a true hero, you have to know who you are. You have to have a strong grip on reality. But this very reality is often questioned, with a common argument being that we live in an irrational world, and that the existence of the so-called “irrational numbers” in nature make any attempt to understand reality impossible.

How can we respond to such a claim?
In this adapted excerpt from the book Philosophy for Heroes: Knowledge, author Clemens Lode sheds light on how the source of these numbers is indeed quite rational.

   Nature’s Irrational Golden Ratio

An adapted excerpt from Philosophy for Heroes: Knowledge
The term “irrational” suggests that there is something in the universe located outside of our perception and our minds. Irrational numbers are infinite and uncountable since they, unlike rational numbers, cannot be constructed from natural numbers in a finite number of steps.

Irrational numbers do not refer to quantities or ratios and do not appear in nature as such. Instead, they refer to processes.

The fundamental problem that a plant must overcome during its growth process is to get as much sunlight as possible on its leaves. If the leaves are arranged according to a regular (i.e., rational) pattern, such as leaf / quarter-turn / leaf / quarter-turn, … the leaves will eventually overshadow one another. The solution is to find an angle of rotation which can be continuously repeated so that no leaf grows directly above another (see Figure 1).

Figure 1:
Optimal leaf arrangement

Nature's solution is the so-called golden ratio, which can be calculated to arbitrarily high accuracy. In the case of plants, this number sequence is generated through cell division with a simple rule: Each mature cell divides and new cells require a certain length of time to mature. The process would occur as follows:
  • From the new cell develops a mature cell. [1 cell]
  • From the mature cell develops a mature cell and a new cell. [2 cells]
  • From the mature cell develops a mature cell and a new cell, and from the new cell develops a mature cell. [3 cells]
  • From the two mature cells develops two mature cells and two new cells, and from the new cell develops a mature cell. [5 cells]
  • From the three mature cells develops three mature cells and three new cells, and from the two new cells develop two mature cells. [8 cells]
  • And on and on...
As this sequence propagates, a special sequence is generated: 1, 2, 3, 5, 8, 13, 21, 34, etc., whereby the ratio of two successive numbers gives an increasingly accurate value of the golden ratio (1.618033…):

3/2 = 1.5
5/3 = 1.66…
8/5 = 1.6
13/8 = 1.625
21/13 = 1.6153…

If we calculate a complete rotation based on the golden ratio, we obtain a series of angle values that minimize overlaps between leaves and thus maximize the amount of sunlight the plant can absorb. This golden ratio can be commonly found throughout nature (including human bodies). In fact, our own perception is calibrated to find objects displaying this ratio to be particularly appealing.

If we draw squares using this number sequence as the lengths of sides, we obtain the “golden spiral” (see Figure 2), which can be found in snails and flowers (see Figure 3).

Figure 2: Golden spiral

Figure 3: Golden spiral in a chamomille flower

We can conclude that irrational numbers do not represent quantities of elements but instead are indicative of a very specific infinite process.

The complexity of reality stems from the fact that it is a product of infinitely repeating processes.

The world is complex. Yet it is also comprehensible. We can, with the right skills and perspective, achieve clarity, even about aspects of reality that seem “irrational.” 


"Heroism" is more than a single act. It's a way of life.

Challenge the traditional idea of "the hero" and discover your own story.

Print and preview available on (or, Kindle ebook available at, too.

The German edition is coming soon.
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