These are cardstock designs, so if you laser cut the pieces they will all stay attached to each other.
And if you are interested, here’s how you make the pieces.
Triangle to Square:
This transformation was made based on Ernest Irving Freese’s design.
Square to Hexagon:
Horizontal lines across the hexagons, a square array tilted at an angle. Then you superimpose them and move them around till you find the least number of pieces in a parallelogram formed by the lines. Parallelogram can reform into a square, and all the pieces fit together to remake the Hexagon. Just beside the hexagon, and squares in the grid have the same area.
Excerpt from a book: Harry Lindgren’s book, Geometric Dissections
Found in this article by Frederickson: https://www.cs.purdue.edu/homes/gnf/book4/Booknews4/ch9.html
Resulting in this figure.