The secret of the 7th row - visually explained
Mathologer・2 minutes read
When stacking circles in a box, the seventh row always remains perfectly level due to the symmetric structure of the stack. The tip of a pyramid built with circles is always equidistant from the walls, displaying half-turn symmetry and a visual proof of the parallel grid segments.
Insights
- Moving the two inner circles of the first row causes subsequent rows to be crooked, except for the seventh row, which remains perfectly level due to the symmetric structure of the stack, showcasing half-turn symmetry.
- Building a pyramid with circles results in the tip always being equidistant from the walls, with the top circle of the stack being at the center and exhibiting half-turn symmetry, demonstrating the parallel nature of the grid segments and the importance of maintaining specific gap ranges between adjacent circles for the stack to behave correctly.
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Recent questions
How are circles stacked in a box?
Circles are stacked row by row, with the first row level and equally spaced. Moving inner circles in the first row causes subsequent rows to be crooked, except for the seventh row, which remains level.
What ensures a level row in circle stacking?
A magic number ensures a level row regardless of the initial circle count. Starting with six circles at the bottom, row 11 remains level due to this magic number.
What happens when building a pyramid with circles?
Building a pyramid with circles results in the tip always being equidistant from the walls. The top circle exhibits half-turn symmetry and is at the center.
Why does the seventh row remain level in circle stacking?
The seventh row remains level due to the symmetric structure of the stack, with base circles being horizontally aligned, leading to half-turn symmetry.
How does widening the box affect circle stacking?
Widening the box causes the top of the stack to no longer be level, as overlapping circles occur. Gaps between adjacent circles must be within a specific range for the stack to behave correctly.
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Summary
00:00
Circle Stacking Theorem: Symmetry and Stability
- Four equally spaced circles in a box are stacked row by row, with the first row being perfectly level and equally spaced, leading to subsequent rows being the same.
- Moving the two inner circles of the first row causes subsequent rows to be crooked, except for the seventh row, which remains perfectly level.
- Starting with six circles at the bottom, row 11 remains level, with a corresponding magic number ensuring a level row regardless of the initial circle count.
- Building a pyramid with circles results in the tip always being equidistant from the walls, with the top circle of the stack being at the center and exhibiting half-turn symmetry.
- A visual proof explains why the tip of the pyramid is in the middle of the walls, showcasing the parallel nature of the grid segments.
- The seventh row is always level due to the symmetric structure of the stack, with the base circles being horizontally aligned, leading to a half-turn symmetry.
- Widening the box causes the top of the stack to no longer be level, as overlapping circles occur due to gaps between adjacent circles being too wide.
- To maintain the stack's properties, the gap between adjacent circles at the bottom must be within a specific range, ensuring the stack behaves correctly.
- The circle stacking theorem, discovered by Charles Payan in 1989, showcases beautiful properties of circle stacking, with practical applications remaining unknown.
