User:Giftlite

0	j	1
φ	5	e
2	π	3

Giftlite

I expect to pass through this world but once; any good thing therefore that I can do, or any kindness that I can show to any fellow creature, let me do it now; let me not defer or neglect it, for I shall not pass this way again.

Etienne De Grellet

It's my delight to share with you my discovery of a new type of 4x4 magic squares in which the numbers in the first row are concatenated to form digits of an identifiable number. If the four numbers in the first row don't add up to 34, we cannot construct a 'normal' magic square. If you see an error, please let me know.

3.	14	15	9
1	17	11	12
24	8	5	4
13	2	10	16
3.	1	41	5
21	16	3	10
15	31	0	4
11	2	6	31
3.	14	1	59
37	34	2	4
30	9	32	6
7	20	42	8
3.	14	15	92
55	51	12	6
53	7	54	10
13	52	43	16
12	98	88	16
46	42	52	74
66	56	64	28
90	18	10	96

{\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots ={\frac {2}{\pi }}

(Viète 1592)

{\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots ={\frac {\pi }{2}}

(Wallis 1655)

\lim _{n\to \infty }\left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)={\frac {\pi ^{2}}{6}}

(Euler 1735)

{\frac {2{\sqrt {2}}}{9801}}\sum _{n=0}^{\infty }{\frac {(1103+26390n)\cdot (4n)!}{396^{4n}\cdot (n!)^{4}}}={\frac {1}{\pi }}

(Ramanujan 1914)

\prod _{j=1}^{4}\prod _{k=1}^{4}\left(4\cos ^{2}{\frac {\pi j}{5}}+4\cos ^{2}{\frac {\pi k}{5}}\right)^{\frac {1}{4}}=36

(Temperley, Fisher, and Kasteleyn 1961)

3.	14	15	9
1	17	11	12
24	8	5	4
13	2	10	16
3.	1	41	5
21	16	3	10
15	31	0	4
11	2	6	31
3.	14	1	59
37	34	2	4
30	9	32	6
7	20	42	8
3.	14	15	92
55	51	12	6
53	7	54	10
13	52	43	16
12	98	88	16
46	42	52	74
66	56	64	28
90	18	10	96

3.	14	15	9
1	17	11	12
24	8	5	4
13	2	10	16
3.	1	41	5
21	16	3	10
15	31	0	4
11	2	6	31
3.	14	1	59
37	34	2	4
30	9	32	6
7	20	42	8
3.	14	15	92
55	51	12	6
53	7	54	10
13	52	43	16
12	98	88	16
46	42	52	74
66	56	64	28
90	18	10	96

3.	14	15	9
1	17	11	12
24	8	5	4
13	2	10	16
3.	1	41	5
21	16	3	10
15	31	0	4
11	2	6	31
3.	14	1	59
37	34	2	4
30	9	32	6
7	20	42	8
3.	14	15	92
55	51	12	6
53	7	54	10
13	52	43	16
12	98	88	16
46	42	52	74
66	56	64	28
90	18	10	96