I have to admit, I’m not necessarily familiar with the term “tie band.” Bias tape is cut from fabric at a 45° angle to the grainline (hence the name).
It is then
1. folded lengthwise toward the center and ironed, and
2. unfolded again.
3. The ironing has created a crease down the middle. The long edges are folded toward this crease and ironed again.
4. Finally, it’s folded toward the center once more, ironed, and stitched down. This results in a strip that’s 1/4 as wide as the original cut.
Tip: You can make bias tape very easily with a “bias tape maker”—just thread the fabric through, iron it right away, and you’re done; no need for all that folding ;).
Bias tape is also often used to finish open edges—keyword: edge finishing.
It’s best to avoid using ribbons on children’s and baby clothing altogether due to the risk of strangulation. There’s also an EU standard regarding this. When in doubt, always leave them out!
For ties, e.g. on baby hats, stretchy fabrics are often used, for example so-called “jersey noodles.” Cut strips of fabric about 2 cm wide along the grainline, pull the ends apart, let go, and they curl up.
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