The collection

Every essay — page 7

Page 7 of 36, 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

Analysis

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

A disc of radius 0.2 and an ellipse of size 1.22 around the same interval. For 1/(1 + 25x²): the interval from −1 to 1 on the real axis, poles at 0 + 0.2i and 0 − 0.2i, the Taylor disc at 0 of radius 0.20, and the Bernstein ellipse with foci ±1 through the poles, with ρ = 1.2198.

An ellipse, not a disc

Secants closing on the slope of the folium x³ + y³ = 3xy. The curve x³ + y³ = 3xy with secants from (1.001, 0.348) of slopes 1.274, 0.953, 0.830, 0.777, approaching the tangent slope 0.744 given by the equation's partial derivatives.

A slope for a curve that is no function

The slope of x² sin(1/x): discontinuous at nought, and never jumping. The derivative of x^2 sin(1/x) on [−0.12, 0.12], with the value at nought marked and the level 0.5 crossed 77 times.

A slope can swing but never jump

A planimeter's wheel measures an area by going round it. Polar planimeter with arms 2.3 and 2 traced round a closed curve; wheel roll 1.6478, times 2, equals the area 3.2955.

An area measured by walking round it

Divergent and convergent integrands pressing towards each other. x·f(x) on a log scale for 1/x, 1/(x ln x), 1/(x ln x ln ln x) (divergent) and 1/(x (ln x)²), 1/(x ln x (ln ln x)²) (convergent), for x up to 10^300.

No integrand sits on the border

A knot drawn with three sines: 5₂ as a Lissajous knot. Frequencies 2, 3, 7 with phases 0.1, 0.7, 0.3; 7 crossings from above; Alexander polynomial 2t − 3 + 2t⁻¹; determinant 7.

Three sines tie a knot

Four stages of the four-corner Cantor dust. Four panels showing stages 1 to 4 of the four-corner Cantor set: 4, 16, 64 and 256 squares kept at the corners.

A dust that almost every line misses

The Flint Hills series, summed to ten million terms. A staircase of partial sums on a logarithmic scale of n, jumping at 1, 3, 22 and 355 and flat afterwards near 30.3.

A series that waits on π

The Cantor function inside a funnel of exponent log 2/log 3. The Cantor staircase with two curves of the form plus or minus the distance to one quarter raised to the power log 2 over log 3, forming a funnel the staircase stays inside.

The exponent a staircase shares with its set

How many three-state chains constant rates can produce. s 0.05: 94.9%; s 0.1: 90.2%; s 0.2: 79.6%; s 0.3: 70.3%; s 0.5: 47.7%; s 0.7: 22.9%; s 1: 4.3%.

When a table of moves came from steady rates

Where interpolation through 11 nodes magnifies an error. Lebesgue functions of equispaced, Chebyshev and optimal nodes of degree 10; constants 29.900, 2.489, 2.052.

The best nodes have no formula

The factorials, and three smooth curves through all of them. Γ(x+1), Γ(x+1)(1 + 0.5 sin² πx) and the degree-six interpolating polynomial through 0!…6!, on a log scale over 0 ≤ x ≤ 5.4; at x = 1/2: 0.88623, 1.32934, -3.58301.

Only one curve through the factorials bends the right way

6 figures · new

Algebra

Structure — what stays true when the numbers change.

A linear map redrawing the plane. The integer grid before and after a linear transformation; the shaded unit square becomes a parallelogram whose area is the determinant.

A matrix is a picture of what happens to the grid

Multiplying two complex numbers. In the complex plane, multiplying adds the two angles and multiplies the two lengths.

Multiplying is turning

The dot product as a shadow. Two vectors and the shadow the first casts on the second. The shadow is 2.425 long and b is 4.123, so the dot product is 10.000.

The dot product is a shadow

The directions the map leaves alone. Unit vectors and their images under the map. On the two marked lines the image points the same way as the original, stretched by 3.00 and 1.00.

The directions a map leaves alone

Completing the square, as a square. An x by x square with the strip split in half and laid along two sides, leaving a square hole of side 1.5. Filling the hole costs 2.25 and buys a perfect square.

Completing the square, by completing a square

Every relabelling of a 4-gon's corners, and the 8 that are motions. All 24 permutations of the corners drawn one by one, with the 8 that preserve every distance marked; the rest deform the polygon and are not symmetries.

Eight ways to leave a square alone

16 colourings in 6 classes. Every way of colouring the corners, with the ones a motion carries to each other placed on the same row; the number of rows is the number of genuinely different colourings.

Colourings nobody can tell apart

The unit square, mapped: area × 5. The unit square and the parallelogram it becomes under a linear map, with the area of that parallelogram computed from its own corners and set against ad − bc.

The number that says how much room is left

The image of four circles, turning 0 to 3 times. The polynomial applied to circles of four radii, each image drawn as a closed loop with the origin marked, and the number of times the loop goes round it.

A loop that cannot miss the middle

A subgroup of 2, and the 4 blocks it cuts the group into. The 8 symmetries of a 4-gon, split into 4 blocks by composing every element onto the subgroup {e, r²}. The blocks all have 2 elements and no element is in two of them.

The blocks a subgroup cuts out

A whole line arrives at the origin. A linear map whose determinant is zero, drawn before and after. One line of the plane is sent to the origin and the whole plane is sent onto another line; the dimension lost and the dimension kept add to two.

What a map throws away

3 roots, and the two numbers the coefficients already knew. The roots of a degree-3 polynomial, found numerically, with the point they average to. That average, and their product, are readable straight off the coefficients without finding the roots at all.

What the coefficients already know