A 360° clock cannot display time dilation due to varying time scales:

The time scale for proper time (t) and the time scale for time dilation (t') are different. The time scale for proper time (t) is 360°, as represented by the 𝑇(𝑑𝑒𝑔) = 𝑇/360 equation. The time scale for time dilation (t') is greater than 360°, Since, Time scale for Time dilation > 360°. Since the two time scales are not the same, the clock that measures proper time (t) cannot display or measure time dilation (t') in the same units.

As per the Special Theory of Relaitivity, time dilation results from relativistic effects that cause time to run differently for the moving object compared to an observer at rest. Due to this difference, the time dilation cannot be directly measured using the same time scale (clock) used to measure proper time.

Conclusion: proper time (t) and time dilation (t') are associated with different time scales, and a clock that measures proper time cannot directly display or measure time dilation in the same units. The relativistic effects of time dilation cause time to be dilated or stretched for the moving object compared to an observer at rest, leading to different time scales for proper time and time dilation.

Mathematical Representation:

The equation of time dilation is 𝑡՚ = 𝑡/√(1 − 𝑣²/𝑐²) where 𝑡′ is dilated time, 𝑡 is proper time, v is relative speed, and c is the speed of light in free space.

1. 𝑇(𝑑𝑒𝑔) = 𝑇/360 = (1/𝑓)/360 = ∆t; Time scale = 360 (𝑇/360); t < t';  
2. Time scale for Proper time = 360°; Proper time = t; 
3. Time scale for Time dilation > 360°; Time dilation = t';  
4. Since, [Time scale for Proper time]  ≠  [Time scale for Time dilation];
5. Therefore, Time scale (clock) for Proper time cannot display Time dilation.