TITLE: Construct homotopy from $(\alpha \cdot \beta) \cdot \gamma$ to $\alpha \cdot (\beta \cdot \gamma)$ explicitly
QUESTION [5 upvotes]: I understand the general idea of a homotopy, but I'm a little lost on how to create them myself. For example if I wanted to show 
$$ \text{Let} \: \alpha, \beta, \text{and} \: \gamma \: \text{be paths} \: I \to X, \: \text{from} \: x_{0} \: \text{to} \: y_{0}, y_{0} \: \text{to} \: z_{0}, \: \text{and}  \: z_{0} \: \text{to} \: u_{0}. \: \text{Then} \: \\
(\alpha \cdot \beta) \cdot \gamma \sim \alpha \cdot (\beta \cdot \gamma) $$
A possible homotopy is $F: I \times I \to X$, given by 
$$\\ F(t,s) = 
\begin{cases}
\alpha(\frac{4t}{1+s}) & 0 \leq t \leq \frac{s+1}{4} \\
\beta(4t-1-s) & \frac{s+1}{4} \leq t \leq \frac{s+2}{4} \\
\gamma(\frac{4t - 2 - s}{2-s}) & \frac{s+2}{4} \leq t \leq 1 \\
\end{cases}$$
What I don't understand is where this comes from. What is the intuition here and how can I form explicit homotopies like this?

REPLY [2 votes]: I have checked the definition and all axioms of the first homotopy group in this note and in this I also explain the exact homotopy you describe. I hope it helps you...