Question 1
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Key Idea
A tiling covers a surface completely without gaps or overlaps.
Geometry Quiz
Space, shape, and structure.
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Question 2
In a right triangle, the square of the longest side equals the sum of the squares of the other two sides.
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Key Idea
The Pythagorean theorem lets you spot right triangles anywhere: if three lengths satisfy $a^2+b^2=c^2$, the angle opposite $c$ is exactly $90^\circ$.
Question 3
What tries to place shapes so they do not overlap?
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Key Idea
In circle packing, the big question is density: what fraction of the plane can circles cover without overlap, and the best possible is the familiar hexagon pattern.
Question 4
A variety that cannot be split into two smaller varieties
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Key Idea
Over an algebraically closed field, a variety is irreducible exactly when its coordinate ring has no zero divisors, so geometry and algebra agree that it has only one "piece".
Question 5
All lines with the same direction share one ...
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Key Idea
In the projective plane, adding points at infinity turns parallel lines into intersecting ones, so every direction corresponds to a unique ideal point and together they form the line at infinity.
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