Abstract --- There are parallelepipeds with edge lengths, face diagonal lengths and body diagonal lengths all positive integers. In particular, there is a parallelepiped with edge lengths 271, 106, 103, minor face diagonal lengths 101, 266, 255, major face diagonal lengths 183, 312, 323, and body diagonal lengths 374, 300, 278, 272. Searches for perfect parallelepipeds led to configurations satisfying necessary quadratic diophantine equations but which are not realizable in R^3; realizable configurations satisfy an additional sixth degree inequality. Brute force searches also give primitive perfect parallelepipeds with some some rectangular faces.

Manuscript preprint: [Preprint on ArXive for Math]

Mathematica^TM computations for proofs in the manuscript: [as html] [as *.nb]

Mathematica^TM computations for 12 examples: [as html] [as *.nb]

Mathematica^TM 3-d visualization for example 1 and 5: [as html] [as *.nb]

Summarizing and extending the examples in the above, here is a list of parameters for the perfect parallelepipeds we have found.
Note: when the manuscript above was prepared we unintentionally did not allow perfect rectangles in our search. Below shows a more complete list of perfect parallelepipeds including some with one or two rectangular faces (e.g. #5 and #10) and where face diagonals match an edge (e.g. #5) and body diagonals match a face diagonal (e.g. #5, #23). Additional examples with three oblique angles meeting at a vertex are listed at the end. At least the first two of these were found by Randall Rathbun before we found the examples below.

n	x1	x2	x3	d23	d13	d12	D23	D13	D12	M1	M2	M3	M4
1	271	106	103	101	266	255	183	312	323	272	278	300	374
2	630	365	335	444	595	385	542	815	955	392	606	906	1108
3	647	540	375	285	448	653	885	958	997	748	758	652	1358
4	739	595	342	463	661	774	853	943	1096	864	828	914	1342
5	840	630	565	565	959	1050	1055	1063	1050	1309	1063	959	1387
6	895	686	652	650	973	939	1170	1227	1289	1225	1055	1155	1685
7	925	579	422	431	577	776	917	1317	1334	638	1074	964	1728
8	952	357	340	493	852	875	493	1148	1141	765	1085	1059	1309
9	1081	840	623	497	1206	1369	1393	1288	1369	1734	1232	1146	1792
10	1120	1035	840	969	1400	1525	1617	1400	1525	1967	1481	1481	1967
11	1381	1038	665	1087	1314	1583	1363	1724	1861	1632	1798	1716	2206
12	1535	1110	955	1015	1030	799	1805	2340	2557	676	1626	2032	3282
13	1575	1500	980	1672	1855	2175	1904	1855	2175	2471	2297	2297	2471
14	1634	1582	959	1267	1845	1888	2289	1943	2604	2473	1689	2387	3115
15	1885	1680	1610	1610	2163	2525	2870	2759	2525	3213	2759	2163	3641
16	2115	1915	1420	1215	1657	1720	3145	3199	3650	2332	2124	2718	4826
17	2255	2240	1540	1316	2255	2685	3612	3135	3605	3587	2509	2709	4837
18	2385	2045	780	1985	1965	2542	2375	2955	3644	2342	2942	3256	4144
19	2517	2354	1248	1886	1581	2077	3262	3645	4409	2005	2779	3473	5471
20	2520	2275	1500	2725	1596	3395	2725	3828	3395	2779	4453	2779	4453
21	2674	1942	1281	1537	2807	1764	2909	3115	4328	2625	1617	4051	4933
22	3080	2688	2241	3315	3809	4088	3675	3809	4088	4795	4525	4525	4795
23	3213	1966	1540	1966	3563	2695	2934	3563	4595	3465	2695	4595	5085
24	3257	2386	1251	2675	3188	1887	2713	3766	5389	1794	2652	5338	5720
25	3343	3024	2730	4074	3343	3565	4074	5107	5285	3565	5255	5285	6545
26	3557	3115	2854	1689	4409	3968	5731	4707	5382	6126	3202	4556	7312
27	3920	3805	2310	3805	4550	5313	5015	4550	5609	6237	5313	5609	6491

Some with three oblique angles meeting at a vertex.

• Clifford A. Reiter and Jordan O. Tirrell*, Pursuing the Perfect Parallelepiped, JP Journal of Algebra, Number Theory and Applications, 6 2 (2006) 279-294. [Abstract] [Auxiliary Materials] [Preprint].
• Jordan O. Tirrell* and Clifford A. Reiter, Matrix Generation of the Diophantine Solutions to Sums of 3 ≤ n ≤ 9 Squares that are Square, JP Journal of Algebra, Number Theory and Applications, 8 1 (2007) 69-80. [Abstract] [Preprint].
• Daniel S. D'Argenio* and Clifford A. Reiter, Families of Nearly Perfect Parallelepipeds, JP Journal of Algebra, Number Theory and Applications, 9 1 (2007) 105-111. [Abstract] [Preprint].