{"version":"https://jsonfeed.org/version/1.1","title":"Curvature of the Mind","home_page_url":"https://curvatureofthemind.com/","feed_url":"https://curvatureofthemind.com/feed.json","description":"Computational art — fractals, physics visualizations, and generative geometry by a recreational physicist.","items":[{"id":"https://curvatureofthemind.com/posts/degree-is-a-coordinate/","url":"https://curvatureofthemind.com/posts/degree-is-a-coordinate/","title":"Degree is a coordinate","content_html":"

Take a polynomial and slide its variable: replace xxx by ax+bax + bax+b. Nothing about the graph is destroyed, it is stretched by aaa and slid by bbb, and it seems obvious that a quintic stays a quintic. It is obvious. But the reason is worth extracting, because the same reason will fail spectacularly in a later article when we allow ourselves to divide.

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The map Ta,b:p(x)↦p(ax+b)T_{a,b} : p(x) \\mapsto p(ax+b)Ta,b​:p(x)↦p(ax+b) is linear in ppp. Two polynomials substituted and then added give the same thing as added and then substituted, so Ta,bT_{a,b}Ta,b​ has a matrix on the space of polynomials of degree at most nnn. Getting it is one application of the binomial theorem. Write p=∑jcjxjp = \\sum_j c_j x^jp=∑j​cj​xj; then

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p(ax+b)=∑jcj(ax+b)j=∑jcj∑k=0j(jk)akb j−kxk,p(ax+b) = \\sum_j c_j (ax+b)^j = \\sum_j c_j \\sum_{k=0}^{j} \\binom{j}{k} a^k b^{\\,j-k} x^k,p(ax+b)=∑j​cj​(ax+b)j=∑j​cj​∑k=0j​(kj​)akbj−kxk,

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so the coefficient of xkx^kxk in the image is ∑j≥k(jk)akb j−kcj\\sum_{j \\geq k} \\binom{j}{k} a^k b^{\\,j-k} c_j∑j≥k​(kj​)akbj−kcj​, and the matrix is

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M(a,b)kj=(jk) akb j−k.M(a,b)_{kj} = \\binom{j}{k}\\, a^k b^{\\,j-k}.M(a,b)kj​=(kj​)akbj−k.

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Everything below the diagonal is zero, because a jjj-th power cannot produce an xkx^kxk with k>jk > jk>j. Every entry on the diagonal is (kk)akb0=ak\\binom{k}{k} a^k b^0 = a^k(kk​)akb0=ak.

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interactive · p(x) ↦ p(ax + b) on the monomial basisopen ↗

Interactive demo: p(x) ↦ p(ax + b) on the monomial basis. Requires JavaScript to run — the source is at /interactive/funcspace/substitution.html.

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Drag aaa and bbb and watch three things move together. The bottom-right entry, boxed, is a5a^5a5. It is the only thing the leading coefficient of the image depends on: the top coefficient of p(ax+b)p(ax+b)p(ax+b) is a5c5a^5 c_5a5c5​, with no contribution from any other cjc_jcj​ and no contribution from bbb at all. The determinant is the product of the diagonal, a0a1⋯an=an(n+1)/2a^{0}a^{1}\\cdots a^{n} = a^{n(n+1)/2}a0a1⋯an=an(n+1)/2, which for n=5n = 5n=5 is a15a^{15}a15. And the coefficient bars redistribute wildly while the highest one refuses to vanish.

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So the statement “an affine substitution preserves degree” is the statement “a triangular matrix with no zero on its diagonal is invertible”, plus the observation that the diagonal here is aka^kak and vanishes only when aaa does. Push aaa to zero in the demo. The matrix collapses to a single row, its rank drops to one, the determinant drops to zero, and the quintic becomes the constant p(b)p(b)p(b). That is not degree preservation failing under stress. It is the map no longer being a change of variable, because x↦bx \\mapsto bx↦b is not a change of anything.

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The triangularity says more than degree preservation. A triangular matrix preserves the whole flag of subspaces P0⊂P1⊂⋯⊂Pn\\mathcal{P}_0 \\subset \\mathcal{P}_1 \\subset \\cdots \\subset \\mathcal{P}_nP0​⊂P1​⊂⋯⊂Pn​: if ppp has degree at most 3, so does p(ax+b)p(ax+b)p(ax+b), for every aaa and bbb at once. Degree is which step of the flag you first appear on, and Ta,bT_{a,b}Ta,b​ moves nobody between steps. Sliding and stretching the variable cannot smuggle a cubic into the quadratics.

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There is a decomposition hiding in the matrix that is worth naming, because the next two articles are about its two halves. Scaling and shifting compose:

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Ta,b=Ta,0∘T1,b,M(a,b)=M(a,0) M(1,b),T_{a,b} = T_{a,0} \\circ T_{1,b}, \\qquad M(a,b) = M(a,0)\\, M(1,b),Ta,b​=Ta,0​∘T1,b​,M(a,b)=M(a,0)M(1,b),

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since applying T1,bT_{1,b}T1,b​ and then Ta,0T_{a,0}Ta,0​ sends p(x)p(x)p(x) to p(x+b)p(x+b)p(x+b) to p(ax+b)p(ax+b)p(ax+b). The first factor M(a,0)M(a,0)M(a,0) is diagonal, diag⁡(1,a,a2,…,an)\\operatorname{diag}(1, a, a^2, \\ldots, a^n)diag(1,a,a2,…,an): the monomials are its eigenvectors and aka^kak are its eigenvalues, which is a fact we will lean on hard when we get to log paper. The second factor M(1,b)M(1,b)M(1,b) is Pascal’s triangle, ones on the diagonal, everything interesting strictly above it. It is unipotent: the identity plus something that is nothing but a nilpotent shadow, and that shadow is the derivative.

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A diagonalizable part and a unipotent part, multiplied together. Anyone who has met the Jordan decomposition will recognize the shape of it, and it is not a coincidence that it appears here: the maps x↦ax+bx \\mapsto ax + bx↦ax+b with a≠0a \\neq 0a=0 form a group under composition — the affine group of the line — and MMM is a representation of that group on Pn\\mathcal{P}_nPn​. The scalings are the semisimple part, the translations are the unipotent part, and the semidirect product structure of the group is visible in the matrices.

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The geometric reading is about roots. If rrr is a root of ppp, then p(ar′+b)=0p(a r' + b) = 0p(ar′+b)=0 exactly when ar′+b=rar' + b = rar′+b=r, so the roots of p(ax+b)p(ax+b)p(ax+b) are the roots of ppp pulled back by the same affine map, r′=(r−b)/ar' = (r - b)/ar′=(r−b)/a. There are nnn of them counted with multiplicity, before and after, in the complex plane. Nothing is created, nothing is destroyed, everything is slid and stretched. The affine maps of the line are exactly the transformations that permute the roots of every polynomial without ever losing one — and the reason they cannot lose one, which will only become clear once we have somewhere for a root to be lost to, is that they fix the point at infinity.

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The affine maps in this article are the same affine maps that drive the fractal designer, where each control point is one contractive map x↦Ax+bx \\mapsto Ax + bx↦Ax+b of the plane rather than the line. Same group, one dimension up, and the same triangular bookkeeping. There it is doing something more dramatic than preserving a degree; see what an affine IFS actually is.

","summary":"Substituting ax + b into a polynomial is a linear map with a triangular matrix. The diagonal is the powers of a, which is the whole reason an affine change of variable can never change a degree.","date_published":"2026-07-14T11:00:00.000Z","tags":["Experiments"]},{"id":"https://curvatureofthemind.com/posts/what-an-affine-ifs-actually-is/","url":"https://curvatureofthemind.com/posts/what-an-affine-ifs-actually-is/","title":"What an affine IFS actually is","content_html":"

An iterated function system is a small pile of functions that shrink things, and the surprising claim is that the pile has a shape.

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Each map in the designer is affine, which is the most boring transformation that still does something interesting: a linear part and a shift.

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wi(x)=Aix+bi,Ai∈R2×2,bi∈R2.w_i(x) = A_i x + b_i, \\qquad A_i \\in \\mathbb{R}^{2\\times 2}, \\quad b_i \\in \\mathbb{R}^2.wi​(x)=Ai​x+bi​,Ai​∈R2×2,bi​∈R2.

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The matrix AiA_iAi​ rotates, scales, and skews; the vector bib_ibi​ slides the result somewhere. That is exactly what a control point in the designer is. When you drag a point you are moving bib_ibi​. When you hold Ctrl and turn it you are rotating AiA_iAi​, and Alt and Shift get you the scale and the skew. The gizmo drawn around each point is a picture of AiA_iAi​.

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\"The

Three affine maps, one of them nudged. The maps are the object you edit; the fractal is what falls out.

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The one condition that matters is that each map has to be a contraction: there is some si<1s_i < 1si​<1 with

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∥wi(x)−wi(y)∥≤si∥x−y∥\\lVert w_i(x) - w_i(y) \\rVert \\le s_i \\lVert x - y \\rVert∥wi​(x)−wi​(y)∥≤si​∥x−y∥

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for every pair of points. Applying wiw_iwi​ moves any two points strictly closer together. Everything below depends on that and on nothing else.

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The Hutchinson operator

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Take a compact set SSS in the plane. Apply every map to the whole of it, and union the results:

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W(S)=⋃i=1Nwi(S).W(S) = \\bigcup_{i=1}^{N} w_i(S).W(S)=⋃i=1N​wi​(S).

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That is the Hutchinson operator, and it eats a shape and produces a shape. It is the entire content of an IFS. Everything the designer does is either computing WWW or arranging for something else to compute it for you.

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Here is the good part. WWW is itself a contraction, not on points but on sets, once you measure the distance between two compact sets with the Hausdorff metric. Its contractivity is s=max⁡isis = \\max_i s_is=maxi​si​, still less than one. Banach’s fixed point theorem then does what it always does: there is exactly one compact set AAA with

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W(A)=A,W(A) = A,W(A)=A,

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and starting from any nonempty compact S0S_0S0​ and iterating Sn+1=W(Sn)S_{n+1} = W(S_n)Sn+1​=W(Sn​), the sequence converges to AAA. Not usually. Always. It does not matter whether you start with a square, a circle, or a single dot.

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That set AAA is the attractor, and it is the thing every renderer on the site is trying to draw. It is defined by the maps and by nothing else — not by the starting shape, not by the algorithm, not by the resolution of your screen. Which is precisely why five different renderers have to agree, and why it means something when they don’t.

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Why the picture has holes in it

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The attractor is a fixed point of a set-valued map, and there is no reason for it to be a nice region. Usually it isn’t. If the maps shrink hard enough that their images don’t overlap much, AAA ends up with empty space at every scale, and its area is zero.

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That last part is not a technicality. It is the central practical fact about rendering an IFS, and I spent a while relearning it the hard way. The attractor typically has measure zero: it covers no area at all. Whatever you draw on a pixel grid is not the attractor, it is a picture of a neighbourhood of the attractor, and every algorithm has to decide what to do with a branch of the set that has become thinner than one pixel. The choices are to round it up to a full pixel, round it down to nothing, or carry a fraction. Two of the three are wrong in ways that take a while to notice.

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The collage theorem

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The other half of Barnsley’s idea runs backwards, and it is the reason the designer is a designer and not a viewer. Suppose you have a target shape TTT — a fern, a coastline, your own handwriting — and you want maps whose attractor looks like it. You don’t have to solve for them. You just have to cover TTT with small distorted copies of itself, and then read the maps off the copies.

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Formally, if you can find maps such that W(T)W(T)W(T) is close to TTT, then the attractor is close to TTT too:

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d(T,A)≤d(T,W(T))1−s.d(T, A) \\le \\frac{d\\bigl(T, W(T)\\bigr)}{1 - s}.d(T,A)≤1−sd(T,W(T))​.

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Make the collage error small and the attractor error is small, inflated by 1/(1−s)1/(1-s)1/(1−s). That is the whole game. You never manipulate the fractal directly. You arrange a few affine copies of a shape so they tile the shape, and the fractal follows.

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Which is what you are doing when you drag the control points around: assembling a collage and watching the fixed point of your collage rearrange itself to keep up. The maps are the object. The fractal is a consequence.

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Three ways of computing that consequence follow. The chaos game samples the attractor with a random walk. The collage run backwards iterates WWW on the pixel grid directly. Escape time without the escape never computes WWW at all and asks a different question entirely.

","summary":"A handful of maps that shrink the plane, a theorem that says they have exactly one fixed shape, and a second theorem that tells you how to find the maps for a shape you already want.","date_published":"2026-07-14T09:00:00.000Z","tags":["Experiments"]},{"id":"https://curvatureofthemind.com/posts/what-a-function-does-to-a-space-of-functions/","url":"https://curvatureofthemind.com/posts/what-a-function-does-to-a-space-of-functions/","title":"What a function does to a space of functions","content_html":"

Write down a polynomial of degree at most five and you have written down six numbers. The polynomial 0.1x5−0.55x3+0.9x0.1x^5 - 0.55x^3 + 0.9x0.1x5−0.55x3+0.9x is the list (0,0.9,0,−0.55,0,0.1)(0, 0.9, 0, -0.55, 0, 0.1)(0,0.9,0,−0.55,0,0.1), read off the powers 1,x,x2,…,x51, x, x^2, \\ldots, x^51,x,x2,…,x5, and every polynomial of degree at most five is exactly one such list. Adding two of them adds the lists. Scaling one scales the list. So the polynomials of degree at most nnn are a vector space of dimension n+1n+1n+1, no more mysterious than R6\\mathbb{R}^6R6, and the monomials xkx^kxk are its coordinate axes.

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That reframing is worth more than it looks. A function on the real line is an infinite amount of information, one value per point, and there is no obvious sense in which one function is close to another or built out of others. A point in R6\\mathbb{R}^6R6, on the other hand, is something you already know how to push around. And the things we do to polynomials all day — shift them, stretch them, differentiate them, multiply them by xxx, feed one into another — are maps from that space to itself, or to some other space just like it. They have matrices. You can look at them.

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interactive · five operators on the polynomials of degree ≤ 5open ↗

Interactive demo: five operators on the polynomials of degree ≤ 5. Requires JavaScript to run — the source is at /interactive/funcspace/operator-zoo.html.

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Pick an operator from the dropdown and watch the matrix. Shifting by hhh, sending p(x)p(x)p(x) to p(x+h)p(x+h)p(x+h), fills in an upper triangle whose diagonal is all ones. Scaling, p(x)↦p(ax)p(x) \\mapsto p(ax)p(x)↦p(ax), is diagonal, with aka^kak down the diagonal. The derivative is a single stripe sitting just above the diagonal. Multiplying by xxx is a stripe just below it, and its matrix is not even square — it takes a six-dimensional space into a seven-dimensional one, because it has to. Substituting x2x^2x2 pushes every coefficient twice as far out.

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Every one of those is a linear map of ppp. That is what having a matrix means, and it is a slightly surprising fact, because none of these operations is linear in xxx. Differentiation obeys (p+q)′=p′+q′(p+q)' = p' + q'(p+q)′=p′+q′. Shifting obeys (p+q)(x+h)=p(x+h)+q(x+h)(p+q)(x+h) = p(x+h) + q(x+h)(p+q)(x+h)=p(x+h)+q(x+h). Even substituting x2x^2x2, which does something violent to the graph, does it additively: (p+q)(x2)=p(x2)+q(x2)(p+q)(x^2) = p(x^2) + q(x^2)(p+q)(x2)=p(x2)+q(x2). Linearity in the function and linearity in the variable are different questions, and confusing them is the single most common way to get lost here.

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So if all of them are linear, what actually distinguishes them? Look at the degree readout, not the matrix. The shift and the scale leave the degree exactly where it was. The derivative eats one. Multiplication by xxx adds one. Substituting x2x^2x2 doubles it. The degree is the thing that moves, and it moves in patterns that are much more rigid than the matrices suggest.

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Degree is a strange quantity. It is not linear — deg⁡(p+q)\\deg(p+q)deg(p+q) is usually max⁡(deg⁡p,deg⁡q)\\max(\\deg p, \\deg q)max(degp,degq) but drops without warning when leading terms cancel. It is not continuous: slide the leading coefficient of a quintic to zero and the degree falls off a cliff from 5 to 4 at the last instant. What it is, is a labelling of a nested chain of subspaces,

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P0⊂P1⊂P2⊂⋯⊂Pn,\\mathcal{P}_0 \\subset \\mathcal{P}_1 \\subset \\mathcal{P}_2 \\subset \\cdots \\subset \\mathcal{P}_n,P0​⊂P1​⊂P2​⊂⋯⊂Pn​,

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constants inside linear inside quadratic and so on, each one a hyperplane in the next. Saying deg⁡p=3\\deg p = 3degp=3 says that ppp lives in P3\\mathcal{P}_3P3​ and not in P2\\mathcal{P}_2P2​. That chain is called a flag, and an operator preserves degree precisely when its matrix is triangular in the monomial basis with nothing vanishing on the diagonal. Triangular is not a computational accident. It is what “respects the flag” looks like when you write it down.

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There is a second structure on this space that the vector-space picture completely ignores, and it is the one that makes polynomials interesting rather than merely six-dimensional. You can multiply two of them. You can also compose two of them. Neither operation is available in R6\\mathbb{R}^6R6, neither is linear, and the two of them treat degree in different ways — one adds, the other multiplies. A space with two incompatible multiplications on it and a grading that each one respects differently is a much richer object than a list of numbers, and almost everything that follows is a consequence of those two laws colliding.

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That is the series. First, degree is a coordinate: what the matrix of p(x)↦p(ax+b)p(x) \\mapsto p(ax+b)p(x)↦p(ax+b) looks like, why it is triangular, and why an invertible affine substitution cannot possibly change a polynomial’s degree. Then products add, composition multiplies, which is the pair of laws just mentioned, and which turns out to prove — in one line, with no calculation — that the only polynomials with polynomial inverses are the affine ones.

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Then the two operators that break the symmetry. The derivative is nilpotent, dying after n+1n+1n+1 applications, and its exponential ehDe^{hD}ehD is the shift, which is Taylor’s theorem wearing a matrix. And the projective fix, where we allow (ax+b)/(cx+d)(ax+b)/(cx+d)(ax+b)/(cx+d), watch it tear a pole in the graph, and repair everything by admitting that the missing structure was a point at infinity all along.

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Then log paper and the shape of a law. Change coordinates on the domain instead of acting on the function, and multiplication becomes addition, powers become slopes, and the integer degree we spent five articles protecting relaxes into a real number. That number is a fractal dimension, and it is the reason the rest of this site exists.

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After that, six more articles in which none of the algebra changes and all of the vocabulary does. Give the polynomial two more variables and ask which ones the Laplacian kills. The answer is a space of dimension 2ℓ+12\\ell+12ℓ+1: ℓ\\ellℓ is a degree, that 2ℓ+12\\ell+12ℓ+1 is the number of orbitals in a subshell, and the shapes printed in every chemistry textbook are the sign regions of homogeneous polynomials in xxx, yyy, zzz. The same polynomials, sorted by degree, are the terms of the multipole expansion, each decaying one power faster than the last. And electrostatics, the hydrogen atom, and the quantum oscillator — three different derivations in three different chapters — turn out to be one angular polynomial times three different radial transcendentals, with the potential getting a vote on the second factor only.

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Then what an external field does to that grading, which is most of atomic spectroscopy read as polynomial algebra. An electric field is multiplication by zzz, a harmonic of degree one, and the selection rule Δℓ=±1\\Delta\\ell = \\pm1Δℓ=±1 is nothing but degrees adding. A magnetic field is a diagonal operator whose eigenvalue is the label you had already written on the state. And a crystal field is the lowest-degree polynomial a cube is permitted to have, which turns out to be degree four, which is why the five ddd orbitals split into two and three, and why a ruby is red.

","summary":"A polynomial is a point in a vector space. Shifting, scaling, differentiating, and substituting are maps of that space into itself, and the only question worth asking about any of them is what happens to the degree.","date_published":"2026-07-11T11:00:00.000Z","tags":["Experiments"]},{"id":"https://curvatureofthemind.com/posts/five-ways-to-draw-the-same-fractal/","url":"https://curvatureofthemind.com/posts/five-ways-to-draw-the-same-fractal/","title":"Five ways to draw the same fractal","content_html":"

The fractal designer has been sitting on this site for a while as a single canvas with a single way of drawing on it. It now has five, and a renderer dropdown to pick between them, and a Save PNG button that hands you the frame without the control gizmos painted over it.

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\"The

One set of three affine maps, five renderers, top to bottom: chaos CPU, chaos WebGL2, collage CPU, collage GPU, escape GPU. Every panel is a real export from the designer.

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The five are Chaos CPU, Chaos WebGL2, Collage CPU, Collage GPU, and Escape GPU. They are five genuinely different algorithms, not five styles, and the interesting thing about having them side by side is that they all have to agree. Every one of them is drawing the attractor of the same handful of affine maps. If two of them disagree, at least one is lying, and there is no way to tell which by looking at a single pretty picture in isolation.

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That turned out to matter more than I expected. When I lined the five up and counted lit pixels, three of them disagreed. The chaos game said the attractor covered about 28,700 pixels. Collage CPU said zero, because it faded to black after a few seconds. Collage GPU said 257,741, a filled blob nine times too big. Two renderers, two opposite failures, both of which had been sitting there looking plausible enough on their own.

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The workspace is also no longer square. It was hardcoded to a 900×900 canvas, with the world extent baked to match, so widening the browser stretched the fractal. The viewport now derives its world-space extent from the canvas aspect ratio, and hit-testing scales each axis independently, so you can drag a point across a wide window and it goes where you put it.

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Save PNG turned out to be the easy part. The control gizmos live on a separate overlay canvas stacked on top of the render canvas, which means exporting the render canvas alone already gives you a clean frame. No hiding, no re-render, no second compositing pass. Both WebGL contexts already asked for preserveDrawingBuffer, so the readback works in the GPU modes too. Files come out named for the renderer and the moment, like fractal-chaos-cpu-2026-07-09T03-55-42.png. Every figure in this series was made with it.

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There is also one control that used to exist and now does not. The collage modes had a Hold slider, which multiplied the whole image by 0.997 on every frame. It read like a trail control. What it actually did was decay the attractor — the invariant set, the thing you came to look at — toward zero on every single frame, while clearing nothing, because outside the image of the maps the value is already zero. It could only ever destroy signal. It’s gone.

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What follows is a short series working through the pieces. First, what an affine IFS actually is, which is the shared object all five renderers are chasing and is a good deal simpler than the pictures suggest. Then one article per algorithm: the chaos game, which throws a point around at random and lets a theorem do the work; the collage run backwards, which is the deterministic one and the one that broke worst; and escape time without the escape, which is not really escape time at all but earns its keep anyway.

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Then, because the bugs were more instructive than the features, three ways to lose a fractal: the clipping, the mask, and the feedback loop. Each one produced a picture that looked fine until something else disagreed with it.

","summary":"The fractal designer now carries five renderers, an aspect-aware workspace, and a button that hands you a clean PNG. Three of the five were quietly drawing the wrong thing.","date_published":"2026-07-09T09:00:00.000Z","tags":["Experiments"]},{"id":"https://curvatureofthemind.com/posts/the-logistic-equation-on-a-line/","url":"https://curvatureofthemind.com/posts/the-logistic-equation-on-a-line/","title":"From exponential growth to the logistic curve","content_html":"

Start with one population and the most optimistic thing you can say about it: every member makes new members at a steady clip and nobody ever dies. Then the more of them there are, the faster the whole thing climbs, which written down is just

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x˙=rx,\\dot{x} = r x,x˙=rx,

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and the answer is the exponential x0ertx_0 e^{rt}x0​ert, a number that doubles and doubles and never lets up. Bacteria in a fresh dish do this for an afternoon, money at a fixed interest rate does it forever on paper, and neither one is telling the truth for long, because something always gives. The dish runs out of sugar. So the real question in population biology was never how things grow. It’s what makes them stop.

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One answer is that something eats them. Lotka and Volterra wrote down the simplest version of that back in the 1920s: let the prey xxx breed on their own at rate α\\alphaα, and get eaten at a rate that depends on how often predator runs into prey, which is proportional to the product xyxyxy. The predators yyy do the opposite, starving off on their own at rate γ\\gammaγ but gaining whenever they catch something.

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x˙=αx−βxy,y˙=δxy−γy.\\dot{x} = \\alpha x - \\beta x y, \\qquad \\dot{y} = \\delta x y - \\gamma y.x˙=αx−βxy,y˙​=δxy−γy.

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Now there are two numbers to track, prey and predators, so we’re back to a plane, and the flow on that plane is worth looking at.

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interactive · predator and prey in phase spaceopen ↗

Interactive demo: predator and prey in phase space. Requires JavaScript to run — the source is at /interactive/difeq/predator-prey.html.

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The whole picture is closed loops going around a center, and each loop is a boom and a bust. Follow one around. Prey are plentiful, so the predators feast and multiply. Now there are too many predators, and they eat the prey down faster than it can breed. The prey crashes, the predators are left with nothing and starve, and once they’ve thinned out the prey creeps back up and the whole thing starts over. The two peaks chase each other around the loop, predators always lagging a quarter turn behind the prey they live on. This isn’t a story, it’s in the data. The old Hudson’s Bay Company fur records of hare and lynx pelts swing up and down in exactly this locked, lagging rhythm.

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The dot in the middle, at (γ/δ, α/β)(\\gamma/\\delta,\\ \\alpha/\\beta)(γ/δ, α/β), is the one balance point where births and deaths cancel for both species and nothing changes. But look at how the flow treats it. Nothing spirals in. It’s a center, not a sink, so if you nudge the system off that point it doesn’t come back, it just circles a slightly bigger loop forever. Predators cap the prey, but only by dragging both of them around this endless cycle. Nobody settles down.

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Predators are an outside fix, though. A population will cap itself even with nothing hunting it, just by getting in its own way, competing for the same food and space. That’s the idea Verhulst had in the 1840s, and it’s the cheapest possible brake. Take the runaway exponential and multiply it by the fraction of room still left, (1−x/K)(1 - x/K)(1−x/K), where KKK is the carrying capacity, the crowd the place can actually feed:

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x˙=rx(1−xK).\\dot{x} = r x\\left(1 - \\frac{x}{K}\\right).x˙=rx(1−Kx​).

\n

When xxx is small the bracket is nearly one and the population grows almost exponentially, same as before. As xxx climbs toward KKK the bracket closes toward zero and the growth throttles itself down. Push past KKK and the bracket goes negative and the population falls back. There’s no predator, no second number, just how many there are, so the phase space collapses back onto a single line.

\n
interactive · the logistic equation as a flow on a lineopen ↗

Interactive demo: the logistic equation as a flow on a line. Requires JavaScript to run — the source is at /interactive/difeq/logistic.html.

\n

The streaks are really all riding on the one horizontal line through the middle. I fanned them out vertically so you can see them, but the only thing moving is their left-to-right position. The curve arching over the top is x˙\\dot{x}x˙ itself, the growth rate against how many there are, and it’s an upside-down parabola. Where it crosses zero the flow stops, and those two crossings are the only fixed points the thing has. The red one at x=0x = 0x=0 is extinction, and it’s unstable, because empty the dish almost all the way and the last few multiply and walk away from zero. The blue one at x=Kx = Kx=K is the full dish, stable, with everything on both sides sliding into it. Start below carrying capacity and you rise to it, start above and you fall to it, and either way the trajectory is the same lazy S, quick through the middle where the parabola is tallest, slow at both ends, easing into KKK and staying.

\n

That S-curve is the honest one. The pure exponential was a population that never stops, which nothing does. The predator-prey loops never settle either, and they cost you a whole second species to draw. The logistic gets you the realistic shape, growth that starts fast and then bends over and levels off, out of a single number and one extra term. It’s what actual penned-in populations trace out, reindeer dropped on an island, yeast in a vat of wort, a bloom filling a pond. They don’t rocket to infinity and they don’t oscillate forever. They fill the space they’ve got and stop.

\n

There’s a sting in the tail worth mentioning. Chop time into discrete steps instead of letting it run smoothly, so the population jumps from this year’s count straight to next year’s, and the same tidy equation becomes the logistic map, one of the most famous objects in all of chaos. Same brake, same carrying capacity, but now it can overshoot and correct and overshoot again, and as you turn up rrr it goes from settling calmly, to flipping between two values, to four, to eight, and then into noise that never repeats. The smooth version can’t do any of that. A flow on a line has nowhere to oscillate. To circle back you need a second dimension, the plane the predators and prey were living on, and to truly come apart you need to cut time itself into steps or climb to a third dimension. The equations are barely more than arithmetic. It’s the room they run in that decides what they can do.

","summary":"A single population grows without bound, predators can only cage it into a boom-and-bust cycle, and the honest fix is a population that limits itself: the S-curve.","date_published":"2026-07-06T11:00:00.000Z","tags":["Physics"]},{"id":"https://curvatureofthemind.com/posts/van-der-pol-and-the-birth-of-a-limit-cycle/","url":"https://curvatureofthemind.com/posts/van-der-pol-and-the-birth-of-a-limit-cycle/","title":"Van der Pol and the birth of a limit cycle","content_html":"

The spring and the pendulum both leak energy when you add friction and ring themselves down to a stop. The van der Pol oscillator is the one that fights back. It came out of radio, of all places. Balthasar van der Pol was working on vacuum tube circuits in the 1920s, and a tube can do something a normal resistor can’t, it can act like negative friction, actually pumping energy into the circuit instead of draining it. What he wrote down is a spring whose damping changes its mind depending on how big the swing is:

\n

x¨−μ (1−x2) x˙+x=0.\\ddot{x} - \\mu\\,(1 - x^2)\\,\\dot{x} + x = 0.x¨−μ(1−x2)x˙+x=0.

\n

The whole trick is hiding in that (1−x2)(1 - x^2)(1−x2) out front of the velocity term. When the swing is small, x2x^2x2 is less than one, the factor is positive, and with the minus sign in front it works out to negative damping. Small motions get pumped up instead of worn down. When the swing gets big, x2x^2x2 climbs past one, the factor flips negative, and now it’s ordinary friction hauling the thing back. So tiny wiggles grow and violent ones shrink, and you can already guess that somewhere in the middle there has to be a size that does neither. The knob μ\\muμ sets how hard the circuit pushes. Split it the usual way into position and velocity (x,v)(x, v)(x,v):

\n

x˙=v,v˙=μ (1−x2) v−x.\\dot{x} = v, \\qquad \\dot{v} = \\mu\\,(1 - x^2)\\,v - x.x˙=v,v˙=μ(1−x2)v−x.

\n
interactive · the van der Pol oscillator in phase spaceopen ↗

Interactive demo: the van der Pol oscillator in phase space. Requires JavaScript to run — the source is at /interactive/difeq/vanderpol.html.

\n

Turn μ\\muμ up off zero and watch what the flow does. The center is still a fixed point, the circuit sitting dead with no current, but now it’s a source. Nothing rests there. The negative damping shoves every nearby streak away in an outward spiral. Meanwhile anything you start way out at the edges gets the real friction and spirals inward. Two opposite behaviors, one pushing out from the middle and one pulling in from the rim, and they collide on the bright loop I drew over the flow. That loop is the limit cycle, and it’s a genuinely different animal from anything in the spring or the pendulum.

\n

The spring had a whole nested family of closed loops, one for every energy you cared to give it, and which one you rode depended entirely on how you started. The van der Pol oscillator has exactly one closed loop, and everything in the plane falls onto it. Start small, spiral out to it. Start big, spiral in to it. Start on it, stay on it. The oscillation isn’t something you set up and then watch decay, it’s something the circuit insists on all by itself, and if you knock it off its rhythm it climbs right back. That’s the difference between a bell, which rings only as loud as you hit it and then quiets down, and a heartbeat, which has an amplitude and a period all its own and returns to them after a stumble. Van der Pol actually built early models of the heartbeat out of exactly this equation, which stopped feeling like a coincidence to me once I’d stared at the picture for a while.

\n

Crank μ\\muμ way up and the shape of the loop changes too. At low μ\\muμ it’s a soft ellipse and the oscillation is nearly a clean sine wave. At high μ\\muμ the loop goes lopsided, stretched and cornered, and the motion turns into a relaxation oscillation, long lazy drifts where the state creeps along one edge broken by sudden snaps across to the other. Slow build, fast release, over and over. It’s the rhythm of a dripping faucet or a stick slipping and catching, and it’s the same single curve, just leaned on harder. One factor of (1−x2)(1-x^2)(1−x2) and a tube that pushes back, and you get an oscillator that winds its own clock.

","summary":"An oscillator with friction that changes its mind. Small swings get pumped up, big ones get bled down, and everything meets on a single closed loop the whole plane falls onto.","date_published":"2026-07-06T10:30:00.000Z","tags":["Physics"]},{"id":"https://curvatureofthemind.com/posts/the-pendulum-the-cats-eye-and-a-little-friction/","url":"https://curvatureofthemind.com/posts/the-pendulum-the-cats-eye-and-a-little-friction/","title":"The pendulum, the cat's eye, and a little friction","content_html":"

The pendulum is what you get when you stop lying about the spring. Everybody’s first pendulum in a physics class is secretly a spring in disguise, because the very first thing the textbook does is wave its hands and replace sin⁡θ\\sin\\thetasinθ with θ\\thetaθ. That’s fine for a wristwatch, where the swing never gets big, but it throws away everything interesting. I wanted to keep the sine, so the widget is drawing the honest equation this time.

\n

Here’s where it comes from. Hang a mass on a rigid rod of length LLL and let θ\\thetaθ be the angle off straight down. Gravity pulls the mass straight down with force mgmgmg, but the rod only lets it move along its arc, and the part of gravity that actually pushes it along that arc is mgsin⁡θmg\\sin\\thetamgsinθ. It points back toward the bottom, so once again there’s a minus sign. Balancing that against the mass’s acceleration along the arc and canceling a length gives

\n

θ¨=−gLsin⁡θ.\\ddot{\\theta} = -\\tfrac{g}{L}\\sin\\theta.θ¨=−Lg​sinθ.

\n

Same shape as the spring, restoring force pulling back toward the middle, except the strength of the pull is sin⁡θ\\sin\\thetasinθ instead of θ\\thetaθ. For small swings the two are nearly equal and you’re back to the spring. For big ones they part ways, and near the very top sin⁡θ\\sin\\thetasinθ rolls back to zero, which is the pendulum telling you that balanced perfectly upside down it feels no push at all. Split it into a state pair the same way, angle and angular velocity (θ,ω)(\\theta, \\omega)(θ,ω):

\n

θ˙=ω,ω˙=−gLsin⁡θ.\\dot{\\theta} = \\omega, \\qquad \\dot{\\omega} = -\\tfrac{g}{L}\\sin\\theta.θ˙=ω,ω˙=−Lg​sinθ.

\n
interactive · the pendulum in phase spaceopen ↗

Interactive demo: the pendulum in phase space. Requires JavaScript to run — the source is at /interactive/difeq/pendulum.html.

\n

Leave the damping at zero. Down the middle you get the same nested loops the spring had, because a pendulum making small swings is a spring. But follow them outward and they stop being ellipses. They stretch, flatten along the top, and eventually a special curve comes through and pinches everything into the shape that made me want to build this in the first place, the row of cat’s eyes. The green dots at the bottom of each eye are the stable resting points, the pendulum hanging straight down, and they repeat across the picture because turning the pendulum all the way around, 2π2\\pi2π in angle, brings it right back to where it started. The red dots sitting between the eyes are the pendulum balanced upside down. Those are fixed points too, but the unstable kind, the saddle. Land exactly on one and you’d hang there forever, but the faintest nudge and you’re gone.

\n

The red curve threading through the saddles is the separatrix, and it’s the most important line on the plot. Inside an eye the pendulum is swinging back and forth, over and over, never making it to the top. That’s libration. Outside the eyes, the flow stops looping and streams across the picture in one direction, and that’s the pendulum with enough energy to go over the top and just keep going around and around like a propeller. The separatrix is the exact border between those two lives, the razor’s edge where the pendulum crawls up to the top and arrives with precisely nothing left. Everything the pendulum can do is sorted by which side of that curve you drop it on.

\n

Now bring the damping slider up. Friction bleeds energy the same way it did for the spring, so the closed eyes unwind into spirals and everything drifts down toward the bottom of the nearest well. A pendulum that started out flying over the top loses a little each pass until one time it doesn’t quite clear it, drops into an eye, and rings itself down to rest hanging straight down. On the plot you can watch a streak cross the old separatrix, get captured, and spiral in. The upside-down saddles stay exactly where they were, unwelcoming as ever, because friction doesn’t make balancing on top any easier, it just guarantees you won’t be circling up there for long.

\n

This is also the doorway to the messy stuff. Give the pendulum a push on a timer instead of letting it wind down and that clean sorted picture starts to fold over on itself, and you’re standing at the edge of chaos. I’ve poked at that end of it before. But even the plain damped pendulum, sitting here doing nothing but swinging and stopping, has more geometry packed into it than the spring ever did, and all of it came from refusing to erase one sine.

","summary":"The pendulum is the spring with the small-angle lie removed. Putting the full sine back in bends its phase space into the cat's-eye pattern, and friction drains it.","date_published":"2026-07-06T10:00:00.000Z","tags":["Physics"]},{"id":"https://curvatureofthemind.com/posts/the-spring-equation-and-the-shape-of-simple-motion/","url":"https://curvatureofthemind.com/posts/the-spring-equation-and-the-shape-of-simple-motion/","title":"The spring equation and the shape of simple motion","content_html":"

The spring is the equation everything else in physics gets compared to, so it’s a good place to start looking at phase space. It comes from about the least surprising fact you can state about a spring: the harder you pull it, the harder it pulls back. Stretch it twice as far and it fights you twice as hard. Write that down and you get Hooke’s law, F=−kxF = -kxF=−kx, where xxx is how far the mass has been pushed from its resting spot and kkk measures how stiff the spring is. The minus sign is the whole story. Whichever way you shove the mass, the force points back the other way.

\n

Newton says force is mass times acceleration, and acceleration is the second derivative of position, so the sentence “the force always points back toward the middle” turns into

\n

mx¨=−kx.m\\ddot{x} = -kx.mx¨=−kx.

\n

That’s the spring equation. To draw it I split that one second-order equation into two first-order ones by giving the velocity its own name. Let v=x˙v = \\dot{x}v=x˙. Then the state of the spring at any instant is the pair (x,v)(x, v)(x,v), and the two of them chase each other around:

\n

x˙=v,v˙=−km x.\\dot{x} = v, \\qquad \\dot{v} = -\\tfrac{k}{m}\\,x.x˙=v,v˙=−mk​x.

\n

Position drifts in the direction the velocity points, and velocity gets pulled back toward zero in proportion to position. That pair is exactly the current the widget draws.

\n
interactive · the spring equation in phase spaceopen ↗

Interactive demo: the spring equation in phase space. Requires JavaScript to run — the source is at /interactive/difeq/spring.html.

\n

Leave the damping slider at zero and the flow settles into a set of nested loops going around the origin, all of them closed. Those loops are ellipses, and the reason they close is that a frictionless spring never loses anything. The quantity

\n

E=12v2+12kmx2E = \\tfrac{1}{2}v^2 + \\tfrac{1}{2}\\tfrac{k}{m}x^2E=21​v2+21​mk​x2

\n

is the energy, kinetic plus the energy stored in the stretch, and it doesn’t change as the mass swings. Set it to a constant and you’ve written the equation of an ellipse, which is why I drew a few of them faintly on top of the flow. Each streak is trapped on the loop it started on, forever, tracing the same swing. Turn up the stiffness and the ellipses get taller because a stiffer spring whips the mass through the middle faster, so the same swing carries more speed. The middle point itself never moves. Sit the mass exactly at rest at the center and it stays there, which is why that dot is a fixed point, and because everything nearby just circles it and never falls in, it’s the kind physicists call a center.

\n

Real springs lose energy, so slide the damping up and the picture changes character completely. Now the mass drags against something and the equation grows a term that always opposes the velocity:

\n

x¨=−kmx−cmx˙.\\ddot{x} = -\\tfrac{k}{m}x - \\tfrac{c}{m}\\dot{x}.x¨=−mk​x−mc​x˙.

\n

The closed loops break open. Every streak still circles the origin but each time around it comes back a little closer, and the whole plane turns into a spiral draining into the center. That’s a ringing note fading out, drawn as a picture. Push the damping high enough and the spiral stops being a spiral at all. The mass no longer overshoots, it just leans back to the middle and stops, and the flow slides straight in without a single loop. The crossover between those two behaviors is critical damping, the setting a screen door closer is tuned to so the door swings shut fast without slamming or bouncing.

\n

None of this needs the solution formula. You never have to write down a sine or an exponential to see that a frictionless spring rings forever and a damped one rings down. It’s sitting right there in whether the loops close or spiral, which to me is the nicest argument for looking at these things as flows in the first place.

","summary":"Where the spring equation comes from, and why its phase space is a family of nested ellipses that damping turns into a spiral.","date_published":"2026-07-06T09:30:00.000Z","tags":["Physics"]},{"id":"https://curvatureofthemind.com/posts/drawing-differential-equations-as-wind/","url":"https://curvatureofthemind.com/posts/drawing-differential-equations-as-wind/","title":"Drawing differential equations as wind","content_html":"

Years ago I threw together a little widget for drawing differential equations. It ran on Angular and a pile of jQuery and bootstrap sliders, and like a lot of things from that stretch of my life it quit working the moment one of its dependencies wandered off without it. I’ve been meaning to bring it back for ages. This week I finally sat down and rewrote the whole thing in plain javascript, no framework, nothing left to rot. The original is still sitting on github if you want to see where it started.

\n

The idea behind it I borrowed, more or less, from the wind maps that were floating around back then, the ones that draw the wind over the country as thousands of tiny streaks that comb along with the flow. A differential equation is a flow too. At every point it tells you which way to move and how fast, and if you drop a particle in and let it follow that instruction you trace out a path. The wind map’s trick is to drop in thousands of particles at once and let them all streak along together. Where the flow is calm you get smooth combed lines. Where it swirls or pinches or spirals into a point you can read it straight off the texture, without ever drawing a single arrow.

\n

That texture is really the whole reason I keep coming back to this. A formula for the solution, when you can even get one, tells you about a single starting condition. The picture tells you about all of them at once.

\n

The plane it’s drawing on is what physicists call phase space. The simplest honest example is a mass on a spring. To know what a spring is going to do next it isn’t enough to know where it is, you also have to know how fast it’s moving and in which direction. Those two numbers, position and velocity, are the state. Put position on one axis and velocity on the other and the whole future of the spring becomes a single point sliding around on a plane. The equation is the current of that plane. Here it is:

\n
interactive · the spring equation in phase spaceopen ↗

Interactive demo: the spring equation in phase space. Requires JavaScript to run — the source is at /interactive/difeq/spring.html.

\n

Drag the stiffness up and down and watch the orbits tighten. Add a little damping and the closed loops open into spirals that wind down to the middle. I’ll get into why that particular shape shows up in the next post, but you can already see the thing I find worth the trouble: the equation isn’t a curve on a graph, it’s a weather pattern, and its personality is written in the way the streaks move.

\n

This is the first of a handful of posts where I point the widget at some of the equations everybody meets in a first physics or math course and just look at the shapes. The spring, the pendulum with a bit of friction added back in, the van der Pol oscillator, the logistic equation. In each one I want to start from where the equation actually comes from, some real bit of physics or population counting, and then let the flow show you what that bit of physics feels like. Half of what I learned putting these together was the physics and the other half was how to make it visible. Both halves were worth it.

","summary":"An old widget of mine, rebuilt in plain javascript, that draws the phase space of a differential equation the way those wind maps draw the wind.","date_published":"2026-07-06T09:00:00.000Z","tags":["Physics","Experiments"]},{"id":"https://curvatureofthemind.com/posts/evil-cauliflower/","url":"https://curvatureofthemind.com/posts/evil-cauliflower/","title":"Evil Cauliflower","content_html":"

My current desktop background.\"evil-cauliflower\"

","date_published":"2014-11-25T14:38:52.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/clearing-stuff/","url":"https://curvatureofthemind.com/posts/clearing-stuff/","title":"Clearing some stuff out","content_html":"

\"seasons\"   One thing I like about this composition is the way the texture of the image mirrors and harmonizes with the shape of the entire image. It is complicated and busy, but not overwhelming, so you can still make out the repeated elements that build on each other to make the whole.

","date_published":"2014-11-09T12:48:05.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/new-feature-download-demo-image/","url":"https://curvatureofthemind.com/posts/new-feature-download-demo-image/","title":"New Feature - Download demo image","content_html":"

\"AtomicI’ve been revamping the site, working towards a more visual layout.  Part of that has been going through old posts and attaching an image to every single one of them.  Some of them just needed selecting an image from the content, and some needed an entirely new image generated from the demo code.  In the past, that has meant capturing screenshots or uploading the image to a service like imgur. I figured there has got to be a better way, so with a little bit of simple research, and adding some angularjs to the site, I’ve added a helpful download button to many of the javascript demos on the site.  This includes the 3D Harmonic oscillator demo, and the Atomic Orbital demo as well. While that makes things easier for me, it also makes the site much more useful in general. Now you can pick out an orientation for an atomic orbital and snap a quick shot of it however you want.  Or you can build your own fractal and grab a copy of it. I’ve also gone through the old catalog and updated pages making sure they all still work.  A number of those early projects were one off simple demos.  While neat, I don’t think they’ve lived up to their full potential because of the limits in parameterizing them and putting multiple copies on a single page.  Now that I’m rewriting them using angularjs I’ll be able to integrate them much more richly into my writing. I’m jonesing to take full advantage of much of the code I’ve written.  I’ve only done two basic posts based on a javascript based geodesic solver I’ve written for Schwarzschild black holes.  There is so much more I can do with that.  I haven’t touched on the Doppler effect in the speed of sound demo, or added relativistic effects.

","date_published":"2013-11-28T00:04:58.000Z","tags":["Experiments","Images"]},{"id":"https://curvatureofthemind.com/posts/symmetric-almost/","url":"https://curvatureofthemind.com/posts/symmetric-almost/","title":"Symmetric - almost","content_html":"

\"perfect I like this because it includes a strong two point linear cantor set attractor with a rotation that looks like it should be symmetric, but as your look at the details, the internals are off. Yet, the transformations are so simple the actual symmetries reveal themselves from inspection.

","date_published":"2013-09-21T19:30:34.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/hexagonal-snowflake/","url":"https://curvatureofthemind.com/posts/hexagonal-snowflake/","title":"Hexagonal snowflake","content_html":"

\"hexThis is a tighter version of the last fractal. The same equilateral triangle is in play with tighter compression factors. If you stare at it enough, you can see that the lines are still there, but the inverse grid of whitespace stands out more strongly. As commonly happens in these creations, the lines become obscured with the nodes that are just touching them. It doesn’t take much for our senses to erase the notion of lineness from our vision.

","date_published":"2013-06-17T11:16:29.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/718/","url":"https://curvatureofthemind.com/posts/718/","title":"Hexagonal grid fractal","content_html":"

\"hex This is the last fractal, but symmetrized. The center points are arranged in an equilateral triangle and the scaling factors are all the same. Each transformation is a 180 rotation where the last few have had a 90 degree rotation to give the grid a rectangular structure.

","date_published":"2013-06-14T11:01:57.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/replicated-triangles/","url":"https://curvatureofthemind.com/posts/replicated-triangles/","title":"Replicated triangles","content_html":"

\"crossings\" This one is the same as the last rendering, but I moved the center of the third transform over and got a resonance creating a neat subdivided triangle effect. The repetition around the edges is even more apparent.

","date_published":"2013-06-12T00:24:56.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/713/","url":"https://curvatureofthemind.com/posts/713/","title":"Faded paper","content_html":"

\"paperThe settings for this one are very close to the last.  I’ve just cranked up the scaling factor for the third transform.  I love how this generates a textured effect that reminds me of weathered paper.  The edges still have that same repeated pattern, though it is harder to follow in the center where everything blurs together. One thing I wonder about is if we perceive the amount of information in these images.  While these appear more diffuse and random, they have just as much pattern and rigidness as the more geometric images.

","date_published":"2013-06-11T00:21:21.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/fractal-grid2/","url":"https://curvatureofthemind.com/posts/fractal-grid2/","title":"Fractal Grid2","content_html":"

\"grid2\"The last grid introduced swirls, twists and rotations onto a “rectangular” partial grid. This is chopping parts out and replicating. Again, the simple form lets you easily find the parts that are replicated to make this image. These static images are nice, but nothing helps get a good understanding of how the parts relate like firing up the generator and draging peices around.

","date_published":"2013-06-06T06:08:48.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/fractal-ruler/","url":"https://curvatureofthemind.com/posts/fractal-ruler/","title":"Fractal Ruler","content_html":"

\"ruler\"This one reminded me of a fractal version of the tic marks on a ruler. It’s up there with the minimal spiral for one of my favorite illustrations of the self similarity of fractals. As they become more complex, it becomes more and more difficult to make out the relationships between the different parts. This one being more “one” dimensional seems to make it easier to take in. This one feels like it is constructed out of lines rather than individual points, which seem to make the relationships less mind bending, even though the definitions only differ by a few numbers. The moment it really sunk in that every individual point in the circles I was looking at really really was circular patterns all the way down. I’d been looking at and computing images of various fractals for over 20 years at that point, but had never made that particular connection before. I had to see the parts as an exact copy of the whole instead of each one being subtly different.

","date_published":"2013-06-05T13:08:27.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/grid-deconstructed/","url":"https://curvatureofthemind.com/posts/grid-deconstructed/","title":"Grid deconstructed","content_html":"

\"Grid

","date_published":"2013-03-22T10:22:50.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/another-image-busy-busy/","url":"https://curvatureofthemind.com/posts/another-image-busy-busy/","title":"Another image, busy busy","content_html":"

\"Mess This is part of a series I put together trying to see how far I could go playing with simple symmetries. I’m not going to moan about how I haven’t been posting, but let you know everything I’ve been doing recently. I’ve been moving a number of wordpress sites from apache to nginx, I’m evaluating scholarship applications, preparing several presentations at my day job covering server performance and the migration we are going to do to ASP.NET MVC. On top of all that I just had an intense blast at SXSWi. Here are some of the notes from the panels I went to.

","date_published":"2013-03-17T23:06:32.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/fractal-friday-wisp/","url":"https://curvatureofthemind.com/posts/fractal-friday-wisp/","title":"Fractal Friday - wisp","content_html":"

\"And

","date_published":"2013-03-08T10:00:00.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/fractal-friday-alignment/","url":"https://curvatureofthemind.com/posts/fractal-friday-alignment/","title":"Fractal Friday - alignment","content_html":"

\"awkwardThis image, while not very visually interesting has a lot of accidental symmetry.

","date_published":"2013-03-01T10:00:00.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/fractal-friday-darkness/","url":"https://curvatureofthemind.com/posts/fractal-friday-darkness/","title":"Fractal Friday - darkness","content_html":"

\"darkness

","date_published":"2013-02-22T10:00:00.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/more-crossing-lines-images/","url":"https://curvatureofthemind.com/posts/more-crossing-lines-images/","title":"More crossing lines images.","content_html":"

\"crossing I haven’t had much time to play with the crossing lines. I do like their simplicity and the way they can create complex crossing relationships. I’ll be playing around with braids next, and integrating differential equations and chaos of course.

","date_published":"2013-02-20T13:21:29.000Z","tags":["Experiments"]},{"id":"https://curvatureofthemind.com/posts/fractal-friday/","url":"https://curvatureofthemind.com/posts/fractal-friday/","title":"Fractal Friday","content_html":"

\"does

","date_published":"2013-02-15T10:00:00.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/playing-around-with-overlapping-lines/","url":"https://curvatureofthemind.com/posts/playing-around-with-overlapping-lines/","title":"Playing around with overlapping lines.","content_html":"

\"twist

","date_published":"2013-02-13T12:50:32.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/some-simple-geometric-fractals/","url":"https://curvatureofthemind.com/posts/some-simple-geometric-fractals/","title":"Some simple geometric fractals","content_html":"

These guys use four transformations with simple rotation parameters.  All I changed is the scaling factor of the two central transformations and their centers.  It shows how the nature and feeling of the image can change dramatically with just a few little parameter changes.

\n
","date_published":"2013-02-09T22:30:58.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/just-post-something-jeeezz/","url":"https://curvatureofthemind.com/posts/just-post-something-jeeezz/","title":"Just post something, Jeeezz","content_html":"
\"I

I wanted to make something more linear, but not.

\n

In the mean time, I’ve been playing around with angularjs, plus have been installing laminate floors in my house, and I ran into How to fold a Julia Fractal a few weeks ago. What a fantastic use of 3d at the top of the page, though it doesn’t scroll well on any of my systems. It’s inspired me to rethink my approach for building interactive demos, and the UI of this site. Fun stuff ahead.

","date_published":"2013-01-26T23:01:57.000Z","tags":["Images"]},{"id":"https://curvatureofthemind.com/posts/varying-circle-grid-and-color/","url":"https://curvatureofthemind.com/posts/varying-circle-grid-and-color/","title":"Varying circle grid and color","content_html":"

\"variedPretty intense, yet some unexpected effects.

","date_published":"2013-01-11T14:02:46.000Z","tags":["Experiments","Images"]}]}