The Function x^2 + xy + y^2 – Part 8

8)      On the number table, x² + xy + y² is elliptical in shape, slanted at a 45° angle

On the number table, x² + xy + y² is elliptical in shape, slanted at a 45° angle.  This was alluded to previously when the twelve different integer derivations of the solution 19 formed an elliptical shape on the number table.  It can also be shown on the number table by assigning different shades of colors based on their values.

	-10	-9	-8	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7	8	9	10
10	100	91	84	79	76	75	76	79	84	91	100	111	124	139	156	175	196	219	244	271	300	10
9	91	81	73	67	63	61	61	63	67	73	81	91	103	117	133	151	171	193	217	243	271	9
8	84	73	64	57	52	49	48	49	52	57	64	73	84	97	112	129	148	169	192	217	244	8
7	79	67	57	49	43	39	37	37	39	43	49	57	67	79	93	109	127	147	169	193	219	7
6	76	63	52	43	36	31	28	27	28	31	36	43	52	63	76	91	108	127	148	171	196	6
5	75	61	49	39	31	25	21	19	19	21	25	31	39	49	61	75	91	109	129	151	175	5
4	76	61	48	37	28	21	16	13	12	13	16	21	28	37	48	61	76	93	112	133	156	4
3	79	63	49	37	27	19	13	9	7	7	9	13	19	27	37	49	63	79	97	117	139	3
2	84	67	52	39	28	19	12	7	4	3	4	7	12	19	28	39	52	67	84	103	124	2
1	91	73	57	43	31	21	13	7	3	1	1	3	7	13	21	31	43	57	73	91	111	1
0	100	81	64	49	36	25	16	9	4	1	0	1	4	9	16	25	36	49	64	81	100	0
-1	111	91	73	57	43	31	21	13	7	3	1	1	3	7	13	21	31	43	57	73	91	-1
-2	124	103	84	67	52	39	28	19	12	7	4	3	4	7	12	19	28	39	52	67	84	-2
-3	139	117	97	79	63	49	37	27	19	13	9	7	7	9	13	19	27	37	49	63	79	-3
-4	156	133	112	93	76	61	48	37	28	21	16	13	12	13	16	21	28	37	48	61	76	-4
-5	175	151	129	109	91	75	61	49	39	31	25	21	19	19	21	25	31	39	49	61	75	-5
-6	196	171	148	127	108	91	76	63	52	43	36	31	28	27	28	31	36	43	52	63	76	-6
-7	219	193	169	147	127	109	93	79	67	57	49	43	39	37	37	39	43	49	57	67	79	-7
-8	244	217	192	169	148	129	112	97	84	73	64	57	52	49	48	49	52	57	64	73	84	-8
-9	271	243	217	193	171	151	133	117	103	91	81	73	67	63	61	61	63	67	73	81	91	-9
-10	300	271	244	219	196	175	156	139	124	111	100	91	84	79	76	75	76	79	84	91	100	-10
	-10	-9	-8	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7	8	9	10

Also, substituting x = p – q and y = p + q, we get x² + xy + y² = (p – q)² + (p – q)(p + q) + (p + q)² = p² – 2pq + q² + p² – q² + p² + 2pq + q² = 3p² + q², which is an ellipse without the 45° slant.