Hilbert Curve Infographic

"type": "educational mathematical infographic", "header": "title": "argument name=\"main title\" default=\"Hilbert Curve\"", "subtitle": "argument name=\"degree subtitle\" default=\"Degree 3 (Order 3)\"", "equation": "n = 3 -> 2^n x 2^n = 8 x 8 grid, 2^(2n) = 64 steps" , "layout": "left_panel": "large 2D line plot", "right_sidebar": "stacked informational panels and text blocks" , "main_graph": "axes": "x_axis": "labeled 'x (column index)' with ticks 0 to 7", "y_axis": "labeled 'y (row index)' with ticks 0 to 7" , "grid": "argument name=\"grid size\" default=\"8x8\" dashed light gray lines", "curve": "description": "continuous non-intersecting path filling the grid", "style": "thick line with directional arrows", "colors": "argument name=\"curve colors\" default=\"gradient transitioning through purple, blue, green, yellow, orange, and red\"", "markers": [ "purple dot at bottom left labeled '00'", "blue dot at top left labeled '64'" ], "labels": "various two-digit numbers scattered along the path vertices, such as 01, 02, 32, 65, 70" , "sidebar": "sections": [ "title": "Legend", "type": "box with rounded corners", "count": 4, "items": [ "purple dot: 'Start (step 0)'", "blue dot: 'End (step 63)'", "black arrow: 'Direction of traversal'", "colored lines: 'Curve (colored by sub-quadrant progression)'" ] , "title": "Definition", "type": "text block", "text": "Hilbert curve of degree n maps the unit interval [0,1] onto the unit square [0,1]x[0,1] while preserving locality." , "title": "Properties", "type": "bulleted list", "heading": "For degree 3:", "count": 3, "items": [ "Grid size: 8 x 8 = 2^3 x 2^3", "Total steps: 64 = 2^2*3", "Visits each of the 64 grid points exactly once in a continuous path." ] , "title": "Quadrant structure (recursive)", "type": "2x2 colored grid", "count": 4, "boxes": [ "Top-left blue: 'Q2 (steps 32-63)'", "Top-right yellow: 'Q3 (steps 48-63)'", "Bottom-left purple: 'Q0 (steps 0-31)'", "Bottom-right pink: 'Q1 (steps 16-47)'" ], "footer_text": "Each quadrant is a degree-2 Hilbert curve (recursively defined)." ]