From 3 x 10 or 3 x 11 narrow tubes you should get seven or eight such quadruple secondary wide tubes, each consisting of four primary narrow tubes.
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Step 11: After each such cut across a secondary wide tube you get a parallelogram. The photo below shows a secondary wide tube on the left, and next to it the rectangle resulting from that first cut through one layer of the tube. Subsequent horizontal cuts across such a rectangle, through all half-inch gaps between vertices, create rows. These are on the right side of the photo below. Note how in alternate cut rows the dark triangles point up versus down.
Step 10: With pencil, draw a line (one line) through the middle of that half-inch gap between vertices across each secondary wide tube. Then cut on the line, one layer of the tube only.
On this photo are 24 blocks, each block consisting of four rows of eight triangles. The rows were completed with 63 seams total, and then it took further 24 * 3 = 72 seams to sew the 24*4 = 96 rows into the 24 blocks.
Step 8: Cut the primary wide tube into primary narrow tubes of tube width equal to 2.5 inches (for final square size 2 inches). This is the hardest step: the cuts should be perpendicular to the folds of the tube and at the same time at 45 degrees to the slants in strips. Take time to do this step carefully. The number of narrow tubes that you get depends on the length of the strips; with standard selvedge-to-selvedge strips you should get ten or eleven primary narrow tubes.
Step 7: Press the latest seam.
Step 2: Divide the strips into three groups: each containing seven dark and seven light strips.
Step 9: Sew together the primary narrow tubes into groups of four at a time. Do not match vertices, instead, match seams in adjacent tubes at edges. The first two photos below show such a match from the back, and the photo on the right shows the resulting half-inch gap between vertices on the right side.
Tube piecing projects
by Irena Swanson
Rows of half-square triangles
Step 1: Cut 21 strips of dark fabrics and 21 strips of light fabrics into strips of width 2 + 1/4 + 1/32 inches. Why this strange width number? This is how math works out for the final size of squares to be 2 inches. (For finished size b inches and using quarter inch seam allowance, cut strips at strip width (b + 0.5)/sqrt 2 + 0.5 inches.)
Step 5: Now make an accurate cut at 45 degrees, close to the beginnings of strips. Do take the time to do this accurately.
Step 3: For each group of seven dark and seven light strips, sew them together, alternating dark and light, and offsetting the strips to get approximately angle 45 degrees, as in this photo. At this point the offsetting has to be only approximate, but if you are too far off, you will not be able to make as many half-square triangles.
The same method can also give you individual rows of arbitrary widths, say for making bear's paw, lady of the lake, cut glass dish, (one orientation of) sawtooth border, ...
In addition to the instructions below you may want to look at my three related videos of half-square constructions on YouTube.
To accomplish this slant, I mark on the strip on the left where to line up the strip on the right at sewing time. I have two methods for such marking.
One way is to eyeball and finger-press a 45-degree fold at the right edge in the strip on the left so that the folded top is parallel to the long strip edges but at half an inch away. This is shown in the photo on the left below.
My preferred method is shown on the right below. For that, I lay all the strips with right sides up, lining their tops, and at the same time staggering them by about an eighth of an inch to expose their right edges. Then I draw a line with a pen/pencil in the right margins at 1 + 3/4 + 1/32 inches below the top. (This is half an inch less than the strip width; the ruler has been moved so that the marked line is more visible.)
Step 4: Press the three pre-tubes (any which way, but do press).
Step 12: It remains to rotate half of the rows, mix them up for a random effect, and sew them together four at a time into blocks.
Step 6: Sew the two outside edges together, matching the strips at seam allowance rather than at their edges: you will get a tube. As this is the first appearance of a tube, I call it the primary wide tube.