Sewing the curve
Turn every other apple core a quarter turn, so a bulging side meets a curved-in side. Match the center notches and the corners, pin, and sew with the curved-in side on top, easing it to fit. Sew pieces into rows, then join the rows the same way, matching the corner dots where four pieces meet.
How the template is drawn
All four curves are arcs of one circle size through the corners of a square as wide as the piece. The top and bottom bulge out and the sides curve in by the same amount, so a piece turned a quarter turn fits its neighbor, the way Quilter's Cache sews them: "the concave curve of one core atop the convex curve of another". Tracing a circle and its neighbors, as Patchwork Posse does, gives a height of 1.41 times the width; the button above sets that.
Example: a 5 by 7 in apple core with 1/4 in seams is cut from 5 3/4 by 7 1/2 in rectangles, 6 to a 40 in strip, so 40 pieces take 7 strips (1 5/8 yd with the extra 1/8 yd).
Questions
How do apple cores fit together?
Every other piece is turned a quarter turn, so each bulging side meets a curved-in side of the same curve. That only works when the corners make a square, which is why the width is measured at the corners.
What height should I choose?
A taller piece has deeper curves and a narrower waist. Traced from a circle, the height is 1.41 times the width; a height closer to the width gives gentler curves that are easier to sew.
Do I sew with the concave or the convex piece on top?
The concave (curved-in) side on top, so you can ease it along the bulging side underneath. Pin the center notch and both corners first.
How do I keep the quilt edges straight?
Cut the edge pieces in half across the middle, or trim the finished top straight, leaving a seam allowance beyond the narrowest point.
How do I know the printout is the right size?
Measure the square at the top right of each page: it must be exactly 1 in (or 25 mm). If not, print again at 100% or Actual size, not Fit to page. From a phone, save a PDF and print that at 100%.
Sources: Patchwork Posse, How to make an apple core template; Quilter's Cache, Applecore block.