The collection

Every essay — page 5

Page 5 of 50, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Geometry

Shapes, and the arguments that can be made by rearranging them.

The point on the most connecting lines. two lines 7+7: degrees 8,8,8,8,8,8,8,8,8,8,8,8,8,8; grid 5x5: max 15.

The busiest point in a set

Rounding to the nearest saves three steps in ten. N=10^2: 3.981 / 3.155; N=10^3: 5.893 / 4.483; N=10^4: 7.826 / 5.832; N=10^5: 9.772 / 7.182; N=10^6: 11.702 / 8.520; N=10^7: 13.654 / 9.874; N=10^8: 15.587 / 11.214

Rounding to the nearest remainder

A bracket that shrinks by the golden ratio. 2: [0.00000, 1.00000]; 3: [0.00000, 0.61803]; 4: [0.23607, 0.61803]; 5: [0.23607, 0.47214]; 6: [0.32624, 0.47214]; 7: [0.32624, 0.41641]; 8: [0.36068, 0.41641]; 9: [0.36068, 0.39512]; peak 0.374750.

The search that reuses a point

Each doubling adds a quarter of what the last one added. n=6: A=2.5980762114; n=12: A=3.0000000000; n=24: A=3.1058285412; sliver ratio 0.263305.

The error that predicts itself

How far from equilateral a curved Morley triangle is, shape by shape. First-order coefficient of Morley-triangle spread per unit angle excess on the sphere, over 820 shapes; maximum 0.00303 at angles 23, 19, 139 degrees; zero at the equilateral shape.

The price of curvature in Morley's triangle

The longest perimeter a polygon of diameter one can have. Reinhardt bound 2n sin(pi/2n) for n=3..16 (reached unless n is a power of two), regular n-gon perimeters at diameter 1, and the best quadrilateral found 3.035273 against 2+4 sin 15 = 3.035276.

The longest fence of diameter one

Two sofas for a hallway one unit wide. Hammersley's sofa with exact area 2.20742 and raster area 2.2095; a shape from an optimised Fourier path, raster area 2.2195 with the same raster excess of 0.0021; Gerver's published value 2.2195.

The largest sofa round a corner

Many points on many lines. Grid of 192 points and 128 lines with 512 incidences, against (n m)^(2/3) = 845.

How often points can sit on lines

Eight points that refuse the cube

Lines that cut out as many triangles as possible. Straight-line arrangements found by a seeded climb: 5 triangles from 5 lines and 11 from 7, each meeting the bound.

The triangle a triple point buys

Four perpendiculars from inside, two from outside. Normals to the ellipse x²/4 + y²/1.44 = 1 from (0.55, 0.18) inside its evolute (4) and from (2.3, 0.9) outside it (2), with the evolute drawn.

Four perpendiculars to an ellipse

Four rods, hinged, enclosing the most. Quadrilaterals with sides 3, 4, 5, 6 as one corner angle varies; the largest has area 18.97367, equal to Brahmagupta's √360 = 18.97367, with its corners on a circle of radius 3.2873.

The most four hinged rods can enclose

Langley's adventitious angle. The isosceles triangle with apex 20° and base angles 80°, the cevian from B at 60° to the base meeting AC at D, the cevian from C at 50° meeting AB at E, and the angle BDE, computed as 30° to nine places.

Thirty degrees without a circle

6 figures · new

Analysis

Limits, curves, and what happens when the going does not stop.

A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

A sine wave is a circle seen from the side

8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

Adding up rectangles until they stop being rectangles

Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

A square wave built entirely out of round ones

eˣ and its tangent lines. The exponential curve with tangent lines at several points; at each point the slope equals the height.

The curve that is its own slope

Terms that vanish, a total that does not. The first 24 terms of the harmonic series as bars, with the running total above them. The last bar is 0.042 tall and the total has reached 3.776.

A sum whose terms vanish and whose total does not

Partial sums of sin x. sin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere.

One point's worth of information

Secants closing on the tangent to x². Secant lines through x = 1 and a second point 1.2, 0.8, 0.5, 0.28, 0.12 away, with the slope of each. They approach 2, the derivative there.

The slope of a single point

The same terms, in a different order, adding to whatever is asked

Area is the undoing of slope. Above, a positive function with the area from 0 to 1.80 shaded. Below, that area plotted against where it stops. The lower curve's slope at 1.80 is 1.129, which is exactly the upper curve's height there.

Area is the undoing of slope

7 figures $A square cut into 7 pieces and a remainder. A square divided by cutting off a fixed fraction of what is left, over and over, so that the pieces are the terms of a geometric series and the uncut corner is the tail.$

The sum that fits in one square

Four staircases against a quarter circle, all of length 2. A quarter circle with staircases of 1, 2, 4, 16 steps drawn over it; each hugs the curve more closely than the last and every one of them is exactly 2 long.

The staircase that is not the diagonal