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Home » \[ H \approx 31.8878 \] - Groen Casting

\[ H \approx 31.8878 \] - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 2Question: A mathematician studying algebraic topology explores symmetries of a surface modeled by a regular polygon with rotational symmetry of order $ n $. If the polygon has 12 sides, what is the sum of all distinct rotational symmetries (in degrees) that map the polygon onto itself?

📌 Solution: A regular 12-sided polygon (dodecagon) has rotational symmetry of order 12. The full set of rotational symmetries consists of rotations by multiples of $ \frac{360^\circ}{12} = 30^\circ $. These rotations are: $ 0^\circ, 30^\circ, 60^\circ, \dots, 330^\circ $. This forms an arithmetic sequence with first term $ 0^\circ $, common difference $ 30^\circ $, and 12 terms. The sum of an arithmetic sequence is given by:

📌 S = \frac{n}{2}(a + l) = \frac{12}{2}(0^\circ + 330^\circ) = 6 \cdot 330^\circ = 1980^\circ

📌 Thus, the sum of all distinct rotational symmetries is $ \boxed{1980^\circ} $.

📌 Question: A middle school student is building a robot with gears. One gear rotates 48 times per minute, and another rotates 72 times per minute. If both gears start aligned, after how many minutes will they align again?

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Total colonies = 1,248 + 1,576 + 987 = <<1248+1576+987=3811>>3,811
Dormant colonies = 18% of 3,811 = 0.18 × 3,811 = <<0.18*3811=685.98>>6...
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