I used a keychain skewb and 3x Pyramorphix keychains for the tips, some milliput and some flat plastic.
I used the keychain skewb, and the middles are the fixed parts axles of a skewb, other builders use the corners as the fixed axle.
I justed some spare plastic and milliput and the tips of a Pyramorphix to get the correct shape of the middle pieces and corners.
If I would rebuild it, I would use only the milliput to fill and cover it with some flat plastic. You may use black telephone cards, or do what I did I used the plastic of a Giant 3x3x3 (cost $2.25).
1x keychain Skewb $ 4
3x keychain Pyramorphix keychain $12
1x Milliput bar $10
flat plastic $ 2.25
Sand paper $ 1
I used a keychain skewb, because with a normal skewb the Halpern Pyramid would be 15 cm high.