Miura-ori is the fold used for solar panels on satellites: a sheet of identical parallelograms that collapses in one motion. Geometrically it is rows of zigzags, each shifted alternately left and right. A Loft with Straight faces between the rows makes every facet a flat parallelogram, so the creases stay crisp and catch the light.

A white Miura-ori folded sheet with alternating light and shaded facets

Reference: choose Examples  Miura-ori fold on the canvas toolbar to load the finished definition.

Build it

Points of every row

  1. Place six sliders: Pleats: 16 (4 – 40) and Rows: 12 (2 – 30) with Decimals 0, Pleat width: 6 (2 – 12), Pleat depth: 6 (0 – 10), Row height: 6 (2 – 12) and Shear: 2.5 (0 – 6).

  2. Place two Addition (Maths) components with B 1 after Pleats and Rows, and two Series (Sets) after them: Point no. (0 … 16) and Row no. (0 … 12).

  3. Place a Multiplication (Maths): Point no. → A, Pleat width → B.

  4. Place an Expression (Maths), Row shift, y*(x % 2), with Row no. → x (Graft on x) and Shear → y: odd rows shift right by Shear. One branch per row.

  5. Place an Addition, X, and wire the Multiplication → A and Row shift → B.

  6. Place an Expression, Zigzag, y*(x % 2), with Point no. → x and Pleat depth → y: odd points step back by the depth.

  7. Place a Multiplication, Z: Row no. → A with Graft, Row height → B.

  8. Place a Construct Point (Vector) with X, Zigzag → Y and Z: 13 branches of 17 points.

The sheet

  1. Place a Polyline (Curve) and wire Construct Point → V: one zigzag per row.

  2. Place a Loft (Surface). Wire the polylines → C, tick Flatten on C, and set value St (Straight) to True.

  3. Turn Preview off on Construct Point and Polyline.

The Miura-ori definition on the canvas
Figure 1. The finished definition
The red preview of the Miura-ori sheet
Figure 2. The preview in FreeCAD’s 3D view

What is happening

Each row is the one below translated by (±Shear, 0, Row height). A face between two rows joins a segment of one zigzag to the matching, translated segment of the next, and four points made of a segment and its translate always lie in one plane: a parallelogram. The alternating sign of the shift makes the second set of creases. The geometry tests check that every facet is flat to a millionth of a millimetre.

A normal loft would smooth through the rows and round the creases off; Straight joins each section to the next with ruled faces and keeps them sharp. The sheet updates in about a sixth of a second.

Things to try

  • Set Shear to 0: the sheet becomes a simple pleated fan with creases in one direction only.

  • Set Pleat depth to 0: the sheet goes flat (with the shear still visible as a skew).

  • Raise Shear to 6 for steep, deep folds.

  • Untick St on the Loft and watch the creases melt into waves.