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the one point with nowhere to go a circle on the sphere… …is a circle on the plane the plane runs on past the edge of the drawing
Stereographic projection
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A sphere is a plane plus one point
1 rung · topology
1 1 1 2 2 1 1 3 2 3 3 2 3 1 1 4 2 5 3 5 3 4 4 3 5 3 5 2 4 1 15 fractions, all in lowest terms, none of them twice and left to right they are already in order
Stern brocot
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Every fraction, exactly once
1 rung · number
the Lorenz system at ρ = 28, projected onto the x–z plane past the critical ρ of 24.74: two fixed points, neither attracting, and a trajectory that settles on neither
Strange attractor
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Two lobes and no cycle
1 rung · dynamics
12 lattice points sit on the circle — and 4 × (3 − 0) = 12 divisors of 25: 1, 5, 25 are 1 mod 4, none are 3 mod 4
Sums of two squares
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Two squares, and a lattice
1 rung · number
a b b a abb⁻¹a⁻¹ V 3 − E 2 + F 1 = 2 two-sided a sphere a b a b aba⁻¹b⁻¹ V 1 − E 2 + F 1 = 0 two-sided a torus a b a b abab⁻¹ V 1 − E 2 + F 1 = 0 one-sided a Klein bottle a b a b abab V 2 − E 2 + F 1 = 1 one-sided a projective plane the corners are coloured by the vertex they become, so the count of colours is the V in V − E + F the characteristic and the orientability are computed from the word; the name underneath is looked up from that pair
Surface classification
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Every surface is a sphere with handles
1 rung · topology
-8 -6 -4 -2 2 4 6 8 -2 -1 1 2 x sin x degree 9 is out by 4.7e+0 at x = 5.76
Taylor series
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One point's worth of information
1 rung · analysis
0.5 1 1.5 2 2.5 1 2 3 4 5 x h = 1.2 slope 3.2000 h = 0.8 slope 2.8000 h = 0.5 slope 2.5000 h = 0.28 slope 2.2800 h = 0.12 slope 2.1200
The derivative
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The slope of a single point
1 rung · analysis
-2 -1.5 -1 -0.5 0.5 1 1.5 2 1 2 3 4 5 6 x y height 0.37 slope 0.37 height 1.00 slope 1.00 height 2.72 slope 2.72 height 4.95 slope 4.95
The exponential
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The curve that is its own slope
1 rung · analysis
H centre midpoints of the sides · feet of the altitudes · midpoints up to H all nine sit at 97.0 from the centre, which is half the 194.1 of the circle through the corners checked again on 240 further triangles, every one of which puts its own nine on one circle
Triangle centres
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Nine points on one circle
1 rung · geometry
Voronoi
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The plane, divided by whoever is nearest
1 rung · geometry
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