fedorov · March 13, 2026 21:01
diff --git a/claude_pulleys_gist.txt b/claude_pulleys_gist.txt
 The user is pointing out that the SVG diagram showing the pulley system is not 
 realistic - the rope is intersecting with the pulleys instead of wrapping 
 around them properly. I need to fix the diagram to show a more realistic pulley 
 system where ropes properly wrap around/over the pulleys without intersecting 
 them.
 Let me think about how a block-and-tackle pulley system actually works and how 
 to draw it properly.
 In a typical block-and-tackle system:

 Fixed pulleys are attached to the ceiling
 Moving pulleys are attached to the load
 The rope weaves between fixed and moving pulleys

 For a simple compound pulley with n moving pulleys:

 The rope starts anchored at the ceiling (or fixed block)
 Goes down around the first moving pulley
 Goes up around a fixed pulley
 Goes down around the next moving pulley
 And so on, with the free end being what you pull

 Let me redesign the diagram more carefully. I'll draw proper pulley circles and 
 route the rope tangentially around them so it doesn't intersect. For a single 
 moving pulley with mechanical advantage of 2, the rope anchors at the ceiling, 
 wraps under the moving pulley, then goes back up and over the fixed pulley at 
 the top with the free end hanging down for pulling. For two moving pulleys with 
 mechanical advantage of 4, I'd continue this pattern with additional fixed and 
 moving pulleys alternating.
 The clearest approach is to draw the fixed pulleys in an upper block and the 
 moving pulleys in a lower block, with the rope weaving between them as straight 
 vertical segments tangent to each pulley circle rather than passing through 
 them. to the edge of each pulley circle (not through the center) — the rope 
 arrives on one side and leaves on the other. For a simple MA=2 system with one 
 moving pulley, I'm picturing one end anchored to the ceiling, the rope running 
 under the moving pulley, then up and over the fixed pulley at the top, with the 
 free end hanging down to pull. Let me trace out exactly how that rope path 
 works.
 Writing SVG diagram... simplest approach that looks realistic is:

 Draw pulleys as circles
 Draw rope segments as lines that connect to the TOP or BOTTOM of each pulley 
 (tangent points for vertical ropes)
 Show the rope going AROUND the pulley by drawing a semicircle arc on the 
 appropriate side

 Let me design this more carefully for up to 6 moving pulleys.
 I'll use a block-and-tackle layout with a fixed block at the top containing n 
 pulleys and a moving block at the bottom with n pulleys plus the load. For a 
 mechanical advantage of 2n with n moving pulleys, I need 2n rope segments 
 supporting the lower block, plus one rope end to pull. The cleanest SVG 
 approach is to draw the upper block as a rectangle with small circles inside 
 for the fixed pulleys, and the lower block similarly with the moving pulleys, 
 then connect them with rope segments.
 Now I'm planning the rope routing: the upper block has pulleys at specific 
 x-positions, the lower block has pulleys at corresponding positions, and I'll 
 draw the rope path connecting them with proper tangent points where the rope 
 leaves each pulley. I'm overcomplicating this—let me step back and use a 
 cleaner approach. I'll draw the pulleys as circles and render the ropes as SVG 
 paths that wrap around them tangentially, so the rope connects at the 
 appropriate point depending on whether it's coming from above or below. at the 
 bottom of the circle
 For a standard block and tackle, the rope segments are nearly vertical, so:

 Each rope segment goes from bottom of upper pulley to top of lower pulley (or 
 vice versa)
 The rope wraps: over the top of fixed pulleys, under the bottom of moving 
 pulleys

 Let me try a completely different, simpler layout that's still physically 
 correct:
 For n moving pulleys, draw a "gun tackle" style layout:

 Place pulleys in a single vertical column (alternating fixed on ceiling, moving 
 attached to load)
 Actually this doesn't really work for n > 2

 OK let me just do a simple but clean version:
 I'll draw n moving pulleys stacked vertically, and n fixed pulleys stacked 
 vertically but offset horizontally. The rope snakes between them using arcs to 
 properly wrap around each pulley.
 For a 2-pulley system, I'm picturing fixed pulleys at the top corners and 
 moving pulleys stacked in the middle below them. The rope needs to anchor at 
 one fixed pulley, then weave down and around the moving pulleys, back up to the 
 other fixed pulley—but I keep running into tangles when I try to trace the 
 path. The clearest educational representation is probably a fixed block at the 
 top center with all n pulleys side by side, and a moving block below it with n 
 pulleys also arranged horizontally, connected by 2n+1 rope segments that 
 alternate going down and up. left to down-right alternating between blocks, 
 plus 1 free end)
 The key insight for drawing: alternate ropes going between LEFT side of upper 
 pulley → LEFT side of lower pulley, RIGHT side of lower pulley → RIGHT side of 
 upper pulley, etc.
 Actually let me look at this from a different angle. The user said the rope 
 intersects the pulleys. The fix is to:

 Draw ropes tangent to pulley circles (not through centers)
 Show proper wrap-around

 Let me simplify drastically:
 For n pulleys, I'll draw:

 A upper "fixed" support bar
 n circles below it for fixed pulleys, touching the bar
 A lower "moving" support bar
 n circles above it for moving pulleys
 The rope weaving between them as vertical lines connecting OUTSIDE the circles

 The rope segments (n moving pulleys means 2n rope segments total) should be 
 nearly-vertical lines that go around the outside of each pulley rather than 
 through it. For the positioning, I'll place fixed pulleys at y = fixedY with 
 x-coordinates spaced evenly, and moving pulleys at the same x-positions but 
 lower. The tricky part is getting the rope to actually wrap around the pulleys 
 correctly instead of just connecting their centers—I think I need to connect 
 the left and right edges of the circles in an alternating pattern. But that's 
 getting complicated geometrically. Let me step back and try a clearer schematic 
 approach instead, something that's obviously correct even if not perfectly to 
 scale. , show:

 Ceiling bar
 Fixed pulleys hanging from ceiling (shown as circles with a fixed axle line)
 Moving pulleys (shown as circles with no fixed axle, attached to load block)
 Rope as a path that goes AROUND the pulleys

 The key is the rope path. For rope to wrap around a circle properly:

 If coming from above and going below (rope wraps over pulley): connect at top, 
 arc over, exit at top (other side)
 If coming from below and going above (rope wraps under pulley): connect at 
 bottom, arc under, exit at bottom (other side)

 Wait, that's not right either. For a pulley:

 The rope enters from one side and exits from the other
 For a fixed pulley at top: rope enters from below-left, wraps over top, exits 
 below-right
 For a moving pulley: rope enters from above-left, wraps under bottom, exits 
 above-right

 So the pattern is clear now—at each pulley, the rope enters on one side and 
 exits the other with a semicircular arc between them. Fixed pulleys have the 
 arc on top, moving pulleys have it on the bottom, with straight vertical 
 segments connecting everything together. Now I need to work out the actual 
 coordinates for the rope path, starting with the anchor point at the fixed 
 pulley and tracing through to the load.
 Writing SVG code... needs to be anchored. For MA=2, one end is fixed to the 
 ceiling or the fixed block.
 Let me reconsider the whole setup for MA=2 (1 moving pulley):
 The setup:

 Fixed pulley attached to ceiling at position F (top center)
 Moving pulley attached to load at position M (below F)
 Rope: one end fixed to ceiling (at same height as F, slightly to the side), 
 goes DOWN around M (wraps under from right to left), goes UP and over F (wraps 
 over from left to right), free end hangs down on the right side of F

 So rope path:

 Fixed end at (fx-offset, ceiling_y)
 Down to (left side of M)
 Semicircle under M from left to right
 Up to (left side of F, at equator level)
 Wait, I need to think more carefully about the tangent points where the rope 
 actually touches each pulley. For a rope wrapping around a pulley, it enters 
 and exits tangentially, so I need to figure out exactly where those contact 
 points are based on the rope's direction of approach and the pulley's geometry. 
 ematic:
 The ropes are approximately vertical
 Each rope touches the left or right edge of a pulley (tangent at equator)
 The rope wraps 180° around the pulley

 So for a fixed pulley at (fx, fy) and ropes going down:

 Rope 1 enters at (fx - r, fy) from above (the rope comes from the left side)
 Rope 1 wraps 180° over the top
 Rope 1 exits at (fx + r, fy) going down

 For a moving pulley at (mx, my) and ropes going up:

 Rope enters at (mx - r, my) from below
 Rope wraps 180° under the bottom
 Rope exits at (mx + r, my) going up

 The straight segments between pulleys connect these tangent points.
 Actually, I need to reconsider the geometry here. For the rope to wrap over the 
 top of the fixed pulley, it has to pass above the center, and for it to wrap 
 under the moving pulley, it needs to pass below. The tangent points for 
 vertical ropes are at the left and right edges at the equator, with the arc 
 curving around the top for the fixed pulley and the bottom for the moving one. 
 Let me work through a concrete example with a fixed pulley at (80, 50) and a 
 moving pulley at (80, 150), both with radius 14, to figure out where the rope 
 actually anchors and how it connects. Writing SVG pulley diagram... Writing the 
 diagram layout... I'm realizing the mechanical advantage formula depends on 
 counting the rope segments that actually support the lower block—for n moving 
 pulleys, that's 2n segments total, which means I need to properly arrange how 
 the rope weaves between the upper and lower blocks to create that many 
 supporting segments. Let me work through this more carefully with a concrete 
 example. For a 4:1 block and tackle, I can trace the rope path: it starts 
 fixed, runs down to the first moving pulley, back up over a fixed pulley, down 
 to the second moving pulley, and back up over the final fixed pulley where the 
 free end emerges. This gives me 4 rope segments between the blocks supporting 
 the load, which matches the 4:1 mechanical advantage. Now I need to figure out 
 the x-positions of these vertical segments. eys need to be sized and spaced 
 correctly. If I position the upper and lower pulleys at the same x-coordinates 
 but with a radius of 12, the rope segments between them will naturally be 
 diagonal, which actually works well for a realistic schematic. I'm going to set 
 up the pulley positions so that for n pulleys on each block, they're evenly 
 distributed around a center point.
 Now I'm working through the rope path for a two-pulley system: it starts at a 
 fixed ceiling anchor, runs down to the left side of the lower pulley, wraps 
 underneath it in a semicircle, then exits from the right side and angles upward 
 diagonally. lower[0].x + r
 // Then goes up to... where? It needs to go to one of the upper pulleys.
 // It should go to the left of upper[0] (wrapping over upper[0]):
 //   → but that means the rope went from right of lower[0] up to left of 
 upper[0]
 //   → and those are at the same x (or close) so nearly vertical ✓
 // After wrapping over upper[0], rope exits at RIGHT of upper[0] = upper[0].x + 
 r
 // Then goes down to LEFT of lower[1]:
 //   upper[0].x + r → lower[1].x - r
 //   This is a diagonal rope ✓ (looks realistic!)
 // After wrapping under lower[1], rope exits at RIGHT of lower[1]
 // Then continues up to the left side of upper[1], nearly vertical again, wraps 
 over, and exits right. The free end hangs down for pulling. Now I'm ready to 
 implement this rope path pattern—it alternates between wrapping under the lower 
 pulleys and over the upper ones, creating those diagonal segments between them. 
 Writing SVG arc logic... I'm working through the SVG arc parameters for the 
 rope wrapping under the pulley—it enters at the left equator, dips down through 
 the bottom, and exits at the right equator, which requires a clockwise arc with 
 sweep=1.
 Now I need to figure out the straight rope segments connecting the pulleys. The 
 fixed anchor point should align vertically with the left side of the lower 
 pulley, and I'm realizing the upper and lower pulleys need to share the same 
 x-coordinates for the geometry to work correctly. Looking at how the rope 
 actually connects in these pulley systems—the fixed end ties to the movable 
 block itself, then the rope wraps around the moving pulleys and back over the 
 fixed ones, with the free end being what you pull. For systems with multiple 
 pulleys on each block, this arrangement scales to create the mechanical 
 advantage.
 For the schematic, I'll position the fixed pulleys at the top, the moving 
 pulleys below them, and the load underneath, then route the rope from an anchor 
 point at ceiling level, zigzagging down through the pulley arrangement with the 
 free end extending downward on the right side.
 Setting up the layout now...
 Actually, I'm realizing I need to think through the rope path more 
 carefully—specifically which x-positions each rope segment occupies as it winds 
 through the pulley system, especially for the simple case where both pulleys 
 are vertically aligned. ically:

 Moving pulley at (cx, my)
 Fixed pulley at (cx, fy)

 Rope:

 Segment 1: from anchor at (cx-r, ceilingY) down to the left side of the moving 
 pulley at (cx-r, my)
 Arc around the moving pulley from left to right
 Segment 2: from the right side of the moving pulley at (cx+r, my) straight up 
 to the fixed pulley

 Now I'm working through how the rope wraps around the fixed pulley—it comes up 
 from below on the right side and needs to wrap over the top to exit downward on 
 the left for the free end. The rope enters the fixed pulley from underneath and 
 curves over it to redirect the force properly. Writing SVG arc path... CCW 
 sweep (sweep=0), to (cx-r, cy):

 CCW from (cx+r, cy) means going UP (negative y direction in screen coords), 
 reaching (cx, cy-r) at top, then down to (cx-r, cy)
 Yes! This is the upper arc. ✓

 So for a fixed pulley (rope wraps over top): arc from (cx+r, cy) CCW to (cx-r, 
 cy): A r r 0 0 0 ${cx-r} ${cy}
 For a moving pulley (rope wraps under bottom): arc from (cx-r, cy) CW to (cx+r, 
 cy): A r r 0 0 1 ${cx+r} ${cy}
 Now I'm standardizing the rope paths: moving pulleys have the rope entering 
 from the top-left, wrapping clockwise underneath, and exiting top-right, while 
 fixed pulleys have it entering from bottom-right, wrapping counter-clockwise 
 over the top, and exiting bottom-left. I also need to handle the straight 
 vertical segments connecting the ceiling anchor down to each pulley. I'm 
 tracing through the complete rope path for a mechanical advantage of 2: it goes 
 down the left side of the moving pulley, curves underneath it, travels up the 
 right side to the fixed pulley, arcs over the top, and then hangs freely from 
 the left side of the fixed pulley. The anchor point positioned to the left of 
 the fixed pulley keeps the geometry clean and the rope routing consistent.
 Now I'm extending this to handle multiple pulley pairs with a mechanical 
 advantage of 4, positioning the upper fixed pulleys at different x-coordinates 
 and their corresponding lower moving pulleys directly beneath them.
 With the spacing set to 60 units between pulley pairs, I'm tracing the rope 
 path and discovering a critical issue: when the rope exits the right side of 
 the first moving pulley and needs to reach the left side of the first fixed 
 pulley, they're at different x-positions, which would create a diagonal rope 
 segment — something that shouldn't happen in a proper pulley system where rope 
 segments between pulleys should be vertical or horizontal.
 The real problem is that the pulleys need to be positioned differently. Either 
 the fixed and moving pulleys should be staggered horizontally so the rope can 
 run straight between them, or in a typical block-and-tackle arrangement, the 
 upper and lower blocks have their pulleys offset from each other so the rope 
 can zigzag properly between them without jumping across. -ish ropes, the x 
 positions need to be:

 Rope 1 (leftmost): x = some_x1
 Rope 2: x = some_x2
 Rope 3: x = some_x3
 Rope 4: x = some_x4

 And the pulleys have their left/right equators at these x positions.
 For F0 at x=f0, F1 at x=f1, M0 at x=m0, M1 at x=m1:

 Rope enters left of M0 at x = m0 - r
 Rope exits right of M0 at x = m0 + r
 For this to enter left of F0: m0 + r = f0 - r → f0 = m0 + 2r
 Rope exits right of F0 at x = f0 + r = m0 + 3r
 For this to enter left of M1: m0 + 3r = m1 - r → m1 = m0 + 4r
 For this to enter left of F1: m1 + r = f1 - r → f1 = m1 + 2r = m0 + 6r
 Free end exits right of F1 at x = f1 + r = m0 + 7r

 So the rope segments are at x = m0-r, m0+r, m0+3r, m0+5r, m0+7r. Setting m0-r = 
 8 for the left margin gives m0 = 8+r, and with r=8 and 2 moving pulleys, m0 = 
 16.
 Now I'm working out the general pattern for n pulleys. The rope segments scale 
 linearly with the number of pulleys, so for n pulleys with radius r=8, the 
 rightmost rope position is at 8 + 32n, and I need to account for margins to get 
 the total width needed.
 Looking at the constraints with a 160px viewBox and 8px margins on each side, I 
 have 144px of available space. I should scale the radius inversely with the 
 number of pulleys to keep everything fitting within the viewBox, so larger 
 pulley counts get smaller radii. I'm working through the pattern for n=2 to 
 verify the spacing. With two moving pulleys and two fixed pulleys, the rope 
 traces through five straight segments, and I can map out the x-positions by 
 tracking where each pulley sits relative to the others. The key insight is that 
 adjacent pulleys share boundary points—the right edge of one pulley equals the 
 left edge of the next—which lets me chain the positions together and determine 
 that the total width scales to 8r.
 Now I'm generalizing this to n pulleys of each type, which gives a span of 4nr, 
 and working backward from the constraint that everything needs to fit within 
 the window width minus margins. That means I can solve for the maximum radius 
 allowed for each value of n.
 I'm settling on a formula that picks the radius dynamically based on the number 
 of pulleys, capping it between 6 and 14 pixels. Then I'm positioning the rope 
 attachment points starting from the left margin, spacing them out by twice the 
 radius for each pulley pair. m0 + (2k-1)r for k = 0, 1, ..., 2n
 That's 2n+1 positions (0 to 2n).
 k=0: x = m0-r (left of M0) = leftMargin ✓ if m0 = leftMargin + r
 k=1: x = m0+r = F0 - r → F0 = m0+2r
 k=2: x = m0+3r = F0+r = M1-r → M1 = m0+4r, F0+r=m0+3r ✓
 k=3: x = m0+5r = M1+r=F1-r → F1=m0+6r ✓
 k=2n: x = m0+(4n-1)r = Fn-1+r (right of last fixed pulley = free end attachment)
 Now checking the rightmost position against the canvas bounds: with leftMargin 
 and rightMargin both at 8, the constraint becomes 4n*r ≤ 144, so r ≤ 36/n. The 
 x-centers follow a pattern where moving pulleys are at leftMargin + (4i+1)*r 
 and fixed pulleys continue that spacing offset by 2r.
 Let me verify this with concrete values for n=2 and r=14: the moving pulleys 
 land at 22 and 78, fixed pulleys at 50 and 106, with the free end reaching 120 
 and total width of 128 fitting within the 160-pixel canvas.
 For n=1 with the same radius, I get positions at 22 and 50. With n=6 and r=6, 
 the spacing gets tighter but still fits exactly at 160 pixels total width.
 Now I'm planning out the SVG path that traces the rope through all these moving 
 and fixed pulleys.
 Writing SVG rope path... Writing the arc path... Actually:

 The rope goes from (m0+r, movingY) UP to (m0+r, fixedY) — this is the rope 
 approaching the pulley from below on its left side.
 Then: the rope wraps over the top of the fixed pulley.
 BUT: if the rope arrives at (m0+r, fixedY) going UPWARD, and the pulley center 
 is at (F[0], fixedY) = (m0+2r, fixedY), then the rope is approaching from the 
 LEFT side, going UP.
 This means the contact point is NOT at the equator but somewhere on the 
 upper-left of the pulley...
 For an exact drawing, the tangent from an upward-going rope at x=m0+r to the 
 circle at (m0+2r, fixedY) with radius r: the tangent point IS the equator at 
 (m0+r, fixedY) because the rope at x=m0+r is exactly tangent to the left side 
 of the circle.

 So the rope touches at the equator, wraps counterclockwise over the top of the 
 fixed pulley, then descends to the moving pulley's left side. Checking the 
 x-coordinates: the right edge of the fixed pulley is m0+3r, and the left edge 
 of the moving pulley is also m0+3r, so they connect perfectly at that point.
 The pattern repeats with a zigzag motion: vertical segments alternate with 
 arcs, with the rope descending into each moving pulley, wrapping underneath, 
 then ascending to the next fixed pulley and wrapping over the top. I should 
 re-index using a cleaner coordinate system to track this pattern more 
 systematically. I need to reconsider the rope positioning more carefully. For 
 the first segment when i=0, the rope starts at the ceiling anchor point and 
 goes down to the first moving pulley, but for subsequent segments when i≥1, 
 they start from the right side of the previous fixed pulley at the fixed pulley 
 level rather than the ceiling. Let me trace through where each down segment 
 actually begins and ends. Writing SVG path geometry... the rope touches M[0] 
 tangentially at its left edge and wraps underneath. For the free end, after the 
 rope exits the rightmost fixed pulley at (m0+(4n-1)*r, fixedY), it hangs 
 straight down—but I need to check if this path intersects any of the lower 
 moving pulleys, particularly M[n-1] which is positioned further to the right. 
 ley center. Since the pulley has radius r, its rightmost point is at 
 m0+(4n-3)*r, and the rope at m0+(4n-1)*r is safely 2r to the right of that. ✓
 The design checks out. Now I need to add the anchor point at the ceiling 
 (y=ceilingY, x=m0-r) and draw the lower block as a rectangle connecting all the 
 moving pulley axles with a line extending down to the load.
 For the upper fixed attachment, I'm placing the anchor at the left and the 
 fixed pulleys to its right. The fixed pulleys' axles sit at (m0+(4i+2)*r, 
 fixedY), and I'll connect them with a horizontal bar at that height to 
 represent the fixed support structure. But I should clarify whether the anchor 
 and the pulleys are separate attachment points or part of the same upper block.
 Actually, thinking about how real block and tackle systems work: the upper 
 block contains all the fixed pulleys and attaches to the ceiling as one unit, 
 while the lower block holds the moving pulleys and connects to the load. The 
 rope's fixed end anchors to the upper block frame itself, somewhere between or 
 adjacent to the pulleys. So I need to position the upper block's left edge to 
 include the anchor point, and its right edge to account for where the rope 
 exits.
 Let me just draw the blocks as simple rectangles behind the pulley circles. The 
 upper block spans from the anchor point on the left through all n fixed pulleys 
 on the right, and the lower block similarly spans across all n moving pulleys. 
 I'll position the anchor at the left edge of the upper block and make sure the 
 block dimensions account for the pulley spacing. I'm settling on a cleaner 
 approach: draw the blocks as rectangles with circles for the pulleys inside, 
 weave the rope between them, anchor the upper block on the left side, and have 
 the free end exit from the right going downward. Now I'm writing the code to 
 implement this diagram.
 Setting up the canvas dimensions and calculating pulley positions...
 I need to finalize the color scheme—I'll use CSS variables for the fills since 
 they work in inline SVG within the same document. Now I'm recalculating the 
 vertical spacing to ensure everything fits properly within the viewBox height, 
 starting with the ceiling position at the top. (same, since load block is fixed 
 size)

 Height: 203 + margin ≤ 260 ✓

 Now checking if the free end rope interferes with anything — it exits from the 
 rightmost fixed pulley and drops down past the moving pulleys. The free end is 
 positioned 2r to the right of the rightmost moving pulley, so there's no 
 collision there.
 For the rope's fixed end, I'm realizing the anchor point at the ceiling is 
 directly above the lower block's left edge, which feels awkward. It makes more 
 sense to attach the rope's fixed end directly to the lower block frame instead, 
 like in typical pulley system diagrams. I'm realizing the anchor placement is 
 getting too complicated with the ceiling reference. Let me simplify by drawing 
 the upper block as a rectangle that contains the rope attachment point 
 directly, without needing a separate ceiling anchor—the rope just starts from 
 the left side of the block itself.
 Actually, I think the standard block-and-tackle zigzag layout is the clearest 
 approach here, so I'll stick with that. Now I'm setting up the SVG generation 
 with the canvas dimensions, margin, and pulley radius calculations based on how 
 many pulleys fit.
 Generating SVG code... Writing SVG elements... Writing the SVG elements... 
 Building the rope path... Writing SVG path segments... Adding SVG elements...
 Now I'm verifying the rope path calculations for the pulley system, checking 
 that the anchor and moving pulley positions align correctly when n=1. - Rope 
 down the left side of M[0] at x=8 ✓ (no intersection with M[0] whose center is 
 at x=22, radius 14, leftmost point at x=8 — rope is tangent ✓)

 Arc under M[0] from x=8 to x=36 ✓
 Rope up the right side of M[0] at x=36 and left side of F[0] at x=36 (F[0] 
 center=50, leftmost=36) ✓
 Arc over F[0] from x=36 to x=64 ✓
 Free end at x=64 going down ✓

 Now checking if the free end at x=64 intersects M[0] (center at x=22, radius 
 14, rightmost at x=36) — it doesn't since 64 is well beyond the pulley's 
 rightmost point.
 For n=2, I'm setting up the path with two mirrors and two focal points: M[0] at 
 x=22, M[1] at x=78, F[0] at x=50, F[1] at x=106, starting from anchor x=8. The 
 rope traces down the left side of M[0], arcs underneath it, rises up to arc 
 over F[0], and then continues down toward M[1] at x=64, which matches where the 
 previous segment ended.
 Now continuing with the second mirror: the rope arcs under M[1], rises to arc 
 over F[1], and ends at x=120 heading downward. I'm checking whether this 
 endpoint and the intermediate rope segments properly avoid intersecting the 
 mirror boundaries—the free end at x=120 clears both mirrors, and the vertical 
 segment at x=64 needs verification against the mirror positions. The vertical 
 segment at x=64 is tangent to F[0] on its right side, just grazing it visually, 
 and doesn't intersect F[1] since it's positioned to the left. Now checking the 
 n=3 case with the updated spacing calculations.
 For n=3, I'm tracing through the path construction with the new radii and 
 positions—the rope winds down and around each circle in sequence, and I need to 
 verify the free end at x=152 doesn't exceed the canvas width.
 The margin check confirms everything fits within 160 units, and the tangency 
 conditions hold: the free end clears the rightmost fixed pulley, and the 
 vertical segment at x=104 just touches the right edge of the middle fixed 
 pulley as intended.
 Now I'm working through the arc geometry for the rope going over each fixed 
 pulley—specifically how the arc connects from the left side to the right side 
 of the circle.
 Writing SVG arc paths... I'm verifying that the vertical rope segment on the 
 left side of the first moving pulley doesn't intersect any fixed pulleys—since 
 the rope stays at x=m0-r and the fixed pulleys start further right, there's no 
 collision. Now I need to check the case where the rope descends from the 
 previous fixed pulley and ensure it clears the right edge of the lower block. 
 The free end sits 2r to the right of the rightmost moving pulley and extends 
 down past the lower block without intersecting the load block, which is 
 centered much further left. Now I'm checking the anchor point placement at x = 
 m0-r just below the ceiling—it needs to clear the upper fixed block which 
 starts at m0+r, so there's potential overlap I need to resolve.
 Rather than dealing with overlapping block rectangles, I think I should 
 simplify the whole visual: just draw the ceiling bar, fixed pulleys hanging 
 from it with an axle line, moving pulleys with their own axle line connected to 
 a load bar, the rope weaving between them, a small hook for the anchor on the 
 ceiling, and an arrow marking the free end. I'm also reconsidering the vertical 
 spacing to give everything more breathing room.
 Now I'm working through the y-coordinates: the ceiling bar sits at the top with 
 some height, the fixed pulleys center around y=50 with their axle lines 
 stretching down from the ceiling, the moving pulleys are positioned lower at 
 y=175 to create a good gap between them, and the load bar sits below that. I'm 
 checking that the spacing works well for different pulley radii and that 
 everything fits within the canvas height.
 For the free end of the rope, it needs to extend downward past the load block 
 to around y=225 without intersecting the moving pulleys or load, which is 
 possible since it's positioned to the right of all the main elements. I'm also 
 planning the visual layout: the rope anchor is actually to the left of the 
 fixed pulleys, the ceiling bar spans the full width, and I'll draw axle lines 
 connecting the ceiling down to the fixed pulleys with a small circle marking 
 the anchor point.
 For the load attachment, I'll connect all the moving pulley axles with a 
 horizontal bar and run a single line down from there to the load block, then 
 add a downward arrow on the free end to show where you'd pull it—which for n=1 
 with r=14 would be at x=64.
 Now I'm checking the positioning for different values of n to make sure the 
 free end doesn't intersect with the load block. For n=1, the free end at x=64 
 is well clear of the load centered at x=22. For n=2, the free end would be at 
 x=120, which also clears the load block.
 For n=6, I'm calculating the exact positions: with the moving pulleys spanning 
 from x=14 to x=134, the load block is centered around x=74 and extends from 
 x=52 to x=96. The free end at x=152 is safely beyond that. Now I need to verify 
 the rope path from the free end to the upper pulley doesn't create any visual 
 issues. The free end is exactly tangent to the rightmost fixed pulley's edge at 
 x=152, which confirms the geometry is consistent. Now I'm ready to write the 
 complete code, making sure to use SVG style attributes with CSS variables for 
 text while keeping the schematic elements in hardcoded hex colors that work in 
 both light and dark modes.
 For the rope I'll use mid-gray (#888), the load block in amber (#BA7517) with 
 light text (#FAEEDA), and the pulleys in blue and teal. I need to calculate the 
 free end's bottom position carefully so it extends below the moving pulleys 
 without exceeding the viewBox height — for both pulley sizes this gives me 
 enough room.
 Writing the complete widget... Writing the SVG code... Building the control 
 panel... Building the output cards... Building the interactive visualization... 
 Writing SVG diagram code... Still writing SVG code... Building the rope path... 
 Writing the rope path and legend... Still writing the update function... 
 Configuring the chart options...
 Now I'm verifying the path calculation for the rope segments—the first line 
 should go straight down from the anchor point at the ceiling to the moving 
 pulley position, keeping the same x-coordinate. Continuing through the path, 
 step 4 traces a vertical line up the right edge of M[0] to reach the left edge 
 of F[0], where both pulleys are tangent to the rope. Now I'm checking whether 
 the rope segment from (36, 172) up to (36, 52) properly navigates these 
 constraints without intersecting either pulley's interior. looks correct! Let 
 me also verify n=2:
 r=14, m0=22
 M[0]=22, M[1]=78, F[0]=50, F[1]=106
 Path:

 M 8 18
 i=0:
 L 8 172 (down to left of M[0], x=8)
 A 14 14 0 0 1 36 172 (under M[0])
 L 36 52 (up to left of F[0])
 A 14 14 0 0 0 64 52 (over F[0])
 i=1:
 leftX = 78-14 = 64, rightX = 78+14 = 92, fRightX = 106+14 = 120
 L 64 172 (down from 64,52 to 64,172)
 A 14 14 0 0 1 92 172 (under M[1])
 L 92 52 (up to left of F[1])
 A 14 14 0 0 0 120 52 (over F[1])
 L 120 221 (free end)

 Now I'm verifying that the rope at x=64 satisfies all the tangency constraints: 
 it's tangent to the right edge of F[0], tangent to the left edge of M[1], and 
 properly positioned relative to F[1] and M[0].
 The free end at x=120 is tangent to F[1]'s right edge, exiting at (120, 52) and 
 running straight down. Everything checks out geometrically. But I'm noticing a 
 potential issue with the vertical line at x=8 for n=1 that goes from (8, 18) to 
 (8, 172)—I need to examine whether this creates any conflicts.
 The rope starts at the anchor point (8, 18), which sits at the bottom-left 
 corner of the ceiling bar that spans from x=8 to x=152 at y=12-18. For n=1, the 
 load block is centered at loadCX = 22, and the free end is at x=64, so they 
 don't overlap. For n=2, I'm calculating loadCX = 50 and checking the load block 
 width, which involves some dimension calculations that I need to verify.
 Calculating the load block dimensions...
 I'm reconsidering the load representation — it should just be a simple 
 fixed-width block (44px) hanging from the moving pulley, not a wide span that 
 varies with the rope positions. Let me trace through the path logic to make 
 sure the rope segments are drawing correctly, starting from the anchor point 
 down to the moving block. to (anchorX, movingY). ✓
 The anchor is at the ceiling. Good.
 But wait — the free end of the rope: after the last arc over F[n-1], I'm at 
 (fixedX(n-1)+r, fixedY) = (freeX, fixedY). Then I do L ${freeX} ${freeBot}. So 
 the path continues with a line going DOWN from the fixed pulley level.
 But the path variable at this point ends at (fRightX, fixedY) from the last 
 arc. The L ${freeX} ${freeBot} where freeX = fixedX(n-1)+r = fRightX, so it's 
 just continuing in the same direction. ✓
 Great. Now I realize the path starts at M ${anchorX} ${ceilBot} but then the 
 loop begins with a vertical line down to the moving pulley, and after all the 
 arcs over the fixed pulleys, it continues down to the free end. For n=1, that 
 final segment goes from (64, 52) to (64, 221). ✓
 The path traces correctly. Now I'm ready to write out the complete widget code. 
 Writing SVG arc logic... Writing SVG arc paths... Writing SVG arc paths... 
 Writing the SVG code... The pulleys and ropes are well within 160px. The 
 ceiling bar spans from x=8 to x=152, with the anchor positioned at the leftmost 
 corner (8, 18), which I should make more visible. For n=6, the anchor also 
 lands at x=8, so that checks out.
 Now I'm working out the load width calculation. I need a formula that scales 
 appropriately across different values of n—for n=1 it should be 44, for n=2 
 around 60, and for larger n values it should cap at 60. I'm using Math.max(44, 
 Math.min(60, lbRight - lbLeft)) to handle this range.
 Actually, I'm going to simplify this. The load itself should always be 44 
 pixels wide and centered under the moving block, regardless of the overall 
 mechanism width. Let me just write the code now.
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