THE FOLD / GLITCH / THE BLUE SCREEN / THE MIDPOINT CIRCLE
THE MIDPOINT CIRCLE
draw a circle with integers and 8-fold symmetry
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The midpoint circle algorithm draws a circle on a pixel grid using only integer arithmetic — no floating point, no trigonometry, no square roots — by tracking a decision variable that chooses, at each step, whether to move straight or diagonally inward, always picking the pixel nearest the true circle.
It exploits the circle’s 8-fold symmetry: compute one 45° octant and mirror it into all eight, plotting 8 pixels per step. It is the integer-only companion to Bresenham’s line, and how every early display drew circles.
LIT verified live: over radii 3…40, every plotted pixel lies within 0.5 of the true radius (measured max ~0.49), using integer arithmetic only (window.__midpointcircle). FIG no framing; exact integer geometry.
It exploits the circle’s 8-fold symmetry: compute one 45° octant and mirror it into all eight, plotting 8 pixels per step. It is the integer-only companion to Bresenham’s line, and how every early display drew circles.
LIT verified live: over radii 3…40, every plotted pixel lies within 0.5 of the true radius (measured max ~0.49), using integer arithmetic only (window.__midpointcircle). FIG no framing; exact integer geometry.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-blue-screen — the pixels on the display, beside its sibling the Bresenham line. The midpoint circle is how the screen draws a ring. AVAN (AI) built the instrument: the integer decision variable, the 8-way symmetric plotting, the radial-deviation check.
Credit as content: developed alongside Bresenham’s line work (Jack Bresenham; Michael Pitteway), 1960s–70s. The weave: David names the blue-screen; I walk one octant with an integer decision variable, mirror it eight ways, and confirm every pixel hugs the true circle within half a pixel.
Credit as content: developed alongside Bresenham’s line work (Jack Bresenham; Michael Pitteway), 1960s–70s. The weave: David names the blue-screen; I walk one octant with an integer decision variable, mirror it eight ways, and confirm every pixel hugs the true circle within half a pixel.
3 ONE DIMENSION
The decision variable d: start at 1−r, and at each step either move straight (x−1... no, y+1) or step inward (y+1 and x−1), whichever keeps the pixel nearest the circle — all in integers, no roots.
4 TWO DIMENSIONS · INTERACTIVE
A circle drawn by the algorithm. One octant is walked and mirrored into eight; every pixel stays within half a pixel of the true radius, verified across many radii.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the single octant actually walked, before mirroring.
AVAN’s addition (the inverse-companion): you compute only one eighth of the circle and get the whole for free. The 8-fold symmetry means one octant’s pixels, reflected across the axes and diagonals, reconstruct the entire ring — so the algorithm does 1/8 the work. The inverse of ‘draw all 360°’ is ‘draw 45° and mirror.’ And the decision variable carries the exact rounding error forward (just like Bresenham’s line), so the pixels never drift from the true circle. Magenta is the 7 octants never computed — mirrored for free; green is the single octant actually walked. Symmetry is 8× less work; error-carrying keeps it exact — the same twin virtues as the Bresenham line.
LIT Genuine midpoint/Bresenham circle algorithm (Bresenham, Pitteway, 1960s-70s). Verified live: the integer decision-variable circle (d=1-r, stepping x++/y-- with integer updates) plots 8-way symmetric pixels each within a radial distance of 0.5 of the true radius across r=3..40 (measured max ~0.4894), using integer arithmetic only (window.__midpointcircle.withinHalfPixel).
FIG No framing: the integer decision variable, the 8-way plotting, and the radial-deviation check run in-browser and are exact. The AVAN inverse is honest — only one octant is walked and reflected into eight (1/8 the work), and the decision variable carries the rounding error forward so pixels never drift; magenta is the 7 free-mirrored octants, green the one walked. Ties to the-bresenham.
FIG No framing: the integer decision variable, the 8-way plotting, and the radial-deviation check run in-browser and are exact. The AVAN inverse is honest — only one octant is walked and reflected into eight (1/8 the work), and the decision variable carries the rounding error forward so pixels never drift; magenta is the 7 free-mirrored octants, green the one walked. Ties to the-bresenham.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BLUE SCREEN · David Lee Wise (ROOT0), with AVAN