- Gimbal lock

The loss of one degree of freedom in a multi-dimensional mechanism at certain alignments of the axes. In a three-dimensional three-gimbal mechanism, gimbal lock occurs when the axes of two of the gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space.
When the red and blue gimbals become aligned.

The Gimbal Lock occurs in various forms depending on how it rotates 90 degrees on each axis.
https://en.wikipedia.org/wiki/Gimbal_lock
https://www.youtube.com/watch?v=vHr77Dre25Q
- Euler angle


The middle(green) can transform the innermost(blue),
the outermost(magenta) can transform the two inner(blue, green) rotation axes.
- Euler formula



θ : A real number and is measured in radians.
- The right side of an equation
∴
means the bar which has length 1 is rotated by θ radian from the x-axis.

◦
- If The base of the natural logarithm e has an exponent x-squares :
- x to i :
- if n = 1

- if n=2

- if n=3

- if n=10

- As n increases, the length of the hypotenuse of a right-angled triangle approaches 1.

Therefore
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