Automorphic Numbers

I came across automorphic numbers in a tweet by Cliff Pickover. He pointed out that:

☛ \(376\) is an automorphic number, meaning a number whose square "ends" in the same digits as the number itself. \(376\) has the property that its cube and fourth power also end in the same digits.

• \(376^2=141376\)
• \(376^3=53157376\)
• \(376^4=19987173376\)

I thought I'd investigate how many of these numbers there up to one million. It turns out that there aren't many. They are:

• 0 0
• 1 1 
• 5 25
• 6 36
• 25 625
• 76 5776
• 376 141376
• 625 390625
• 9376 87909376
• 90625 8212890625
• 109376 11963109376
• 890625 793212890625

Surprisingly when we consider the cubes of numbers, the count increases substantially but the same numbers as for the squares reappear:

• 0 0 square also
• 1 1 square also
• 5 125 square also
• 6 216 square also
• 25 15625 square also
• 76 438976 square also
• 376 53157376 square also
• 625 244140625 square also
• 9376 824238309376 square also
• 90625 744293212890625 square also
• 109376 1308477051109376 square also
• 890625 706455230712890625 square also

However, it should be emphasised that there are many other trimorphic numbers apart from these e.g. \(24^2=13824\). The trimorphic numbers are listed in OEIS A033819.

A033819

Trimorphic numbers: \(n^3\) ends with \(n\).

0, 1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, 18751, 31249, 40625, 49999, 50001, 59375, 68751, 81249, 90624, 90625, ...

With fourth powers however, the count again is more modest and the same numbers reappear:

• 0 0 square and cube also
• 1 1 square and cube also
• 5 625 square and cube also
• 6 1296 square and cube also
• 25 390625 square and cube also
• 76 33362176 square and cube also
• 376 19987173376 square and cube also
• 625 152587890625 square and cube also
• 9376 7728058388709376 square and cube also
• 90625 67451572418212890625 square and cube also
• 109376 143115985942139109376 square and cube also
• 890625 629186689853668212890625 square and cube also

This property of these numbers continues indefinitely and as Wikipedia states:

There are four 10-adic fixed points of \( f(x)=x^{2}\), the last 10 digits of which are one of these:

• \( \ldots 0000000000 \)
• \(  \ldots 0000000001 \)
• \(  \ldots 8212890625 \)
  (sequence A018247 in the OEIS)
• \(  \ldots 1787109376 \)
  (sequence A018248 in the OEIS)

Thus we see why all the automorphic number appear as they do, forming OEIS A003226. Apparently such numbers can also be called curious numbers or circular numbers.

A003226

Automorphic numbers: \(m^2\) ends with \(m\).

0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376, 59918212890625, 259918212890625, 740081787109376, ... 

Of course, automorphic numbers can exist in any base. For a given base \(b\), the number of \(b\)-adic fixed points is determined by 2^(number of distinct prime factors). Because 10 is the product of two distinct prime factors, it has \(2^2=4\) fixed points. Likewise with 6 and 12 (even though \(12=2^2 \times 3\), it has only two distinct prime factors). Of course, for prime numbered bases such as 2, 3, 5 etc. and perfect powers such as 4, 8, 9, 16 etc., there are only 2 fixed points and these are the trivial 0 and 1. Here is a permalink that will generate automorphic numbers in any base (up to 36) and for any power.

Applied to base 30 (that is comprised of three prime factors) it can be seen that there are \(2^3=8\) distinct 30-adic fixed points. Here are the 30-morphic numbers up to one million:

• 0 0
• 1 1
• 6 16
• a 3a
• f 7f
• g 8g
• l el
• p kp
• 3a b3a
• 7f 1q7f
• ap 3rap
• j6 c8j6
• mg grmg
• ql nmql
• 13a 1713a
• 2j6 6t2j6
• 3mg e23mg
• q7f mt1q7f
• rap osirap
• sql roisql
• 1q7f 3fe1q7f
• b2j6 42s9b2j6
• csql 5i15csql
• h13a 9k7oh13a
• irap brjsirap
• s3mg qb0fs3mg

There's a lot more to this topic of course but at least this post serves as an introduction to the topic and perhaps I can pursue other aspects later.

Displaying the SageMathCell Interface

Quite some time ago I tried embedding the SageMathCell interface in my blog posts but experienced some glitches that led me to turn to the use of permalinks as an alternative. The permalink is almost as good as having the interface embedded but it does require the user to click a hyperlink. There's something to be said for simple embedding as the user can execute the SageMath code right there on the page without following a hyperlink. In this post I just want to test this method out again, using a sequence from my previous post:

A279093

Numbers that are nontrivially palindromic in three or more consecutive integer bases.

The initial members are:

178, 300, 373, 676, 1111, 1702, 2473, 3448, 4651, 6106, 7837, 9868, 12223, 14926, 18001, 21472, 25363, ...

Pressing evaluate in the embedded SageMathCell interface should produce "the initial members" as listed under the sequence. Now I'll take it a step further and display numbers along with their palindromic equivalents in the three consecutive number bases.

Palindromes In Two Or More Consecutive Number Bases

Having turned a venerable 26457 days old, I searched through the OEIS to find an interesting sequence in which this number appeared. Nothing caught my fancy but I did stumble upon an interesting post to the Mathematics section of StackExchange. See Figure 1.

Figure 1

An extended search within the OEIS, looking for so-called b-files (see my December 2019 article titled In the Groove), had revealed nothing of interest but Google had returned a result because the number appeared in the StackExchange post and that post contained links to the OEIS. The post began as follows:

Lets look at sequences of numbers that are a palindromic number in two consecutive number bases \(b\) and \(𝑏+1\), where \(𝑏 \geq 2\) of course. (And also ignoring trivial one digit palindromes.)

I would conjecture that there are infinitely many numbers palindromic in two consecutive number bases for any two number bases \((𝑏,𝑏+1)\) where \(b \geq 2\). But I do not know how to show that this statement is true.

Turns out, it is in fact not known if the case \((2,3)\) for example, has infinitely many terms, since the OEIS entry for it is written as "...if it exists". There are no clear patterns in this particular sequence, as it seems.

This is the OEIS entry referred to:

A060792

Numbers that are palindromic in bases 2 and 3.

The first few entries are: 0, 1, 6643, 1422773, 5415589, 90396755477, ... so such numbers are relatively sparse. Looking at the first non-trivial number in this sequence (6643), we see that in base 2, 6643 --> 1100111110011 and in base 3, 6643 --> 100010001.

The StackExchange goes on to list numbers that are palindromic is two consecutive number bases (with \(b\) ranging from 2 to 36\)):

Numbers up to 107 in number bases up to 32 (where the ∗ indicates that the number is also palindromic in a third consecutive base) :

(2, 3): 6643, 1422773, 5415589, ... OEIS A060792

(3, 4): 10, 130, 11950, 175850, 749470, 1181729, ... OEIS A097928

(4, 5): 46, 9222, 76449, 193662, 2347506, 2593206, ... OEIS A097929

(5, 6): 67, 98, 104, 651, 2293, 3074, 26691, 27741, 704396, 723296, 755846, 883407, ... OEIS A097930

(6, 7): 92, 135, *178, 185, 5854, 6148, 7703, 186621, 204856, 206620, 213970, 269957, 271721, 279071, ... OEIS A097931

(7, 8): 121, 178, 235, 292, *300, 2997, 6953, 7801, 10658, 13459, 16708, 428585, 431721, 444713, 447849, ... OEIS A099145

(8, 9): 154, 227, 300, *373, 446, 455, 11314, 12547, 17876, 27310, 889435, 894619, 899803, 926371, ... OEIS A099146

(9, 10): 191, 282, 373, 464, 555, 646, 656, 6886, 25752, 27472, 42324, 50605, 626626, 1540451, 1713171, 1721271, 1828281, 1877781, 1885881, 2401042, 2434342, 2442442, 2450542, 3106013, 3114113, 3122213, 3163613, 3171713, 3303033, *3360633, ... OEIS A029965

(10, 11): 232, 343, 454, 565, *676, 787, 898, 909, 26962, 38183, 40504, 49294, 52825, 63936, 75157, 2956592, 2968692, 3262623, 3274723, 3286823, 3298923, 3360633, 3372733, 4348434, 4410144, 4422244, 4581854, 4593954, 5643465, 5655565, 5667665, 5741475, 7280827, 7292927, 8710178, 8722278, 8734378, 8746478, 8758578, 8820288, 8832388, ... OEIS A029966

(11, 12): 277, 410, 543, 676, 809, 942, 1075, 1208, 1220, 38425, 54662, 72351, 75399, 93088, 125430, 1798303, 1817179, 5058385, 5075809, 5093233, 5199361, 5216785, 5550889, 5568313, 5585737, 5603161, 5620585, 7569434, 7727702, 7833830, 7851254, 7868678, 7886102, 9711399, 9728823, 9746247, 

(12, 13): 326, 483, 640, 797, 954, *1111, 1268, 1425, 1582, 1595, 53210, 100636, 104549, 123257, 129198, 151819, 174596, 227806, 8281118, 8305454, 8329790, 8354126, 8502170, 8526506, 9041475, 9065811, 9090147, 9114483, 

(13, 14): 379, 562, 745, 928, 1111, 1294, 1477, 1660, 1843, 2026, 2040, 71905, 105394, 136517, 167458, 170006, 174934, 205875, 208423, 239364, 270487, 342392, 344954, 

(14, 15): 436, 647, 858, 1069, 1280, 1491, *1702, 1913, 2124, 2335, 2546, 2561, 95146, 139667, 181248, 225769, 231874, 267140, 276395, 317766, 454493, 499014, 502179, 

(15, 16): 497, 738, 979, 1220, 1461, 1702, 1943, 2184, 2425, 2666, 2907, 3148, 3164, 123617, 181698, 294260, 348501, 359797, 414038, 472119, 526600, 650217, 708298, 712154, 

(16, 17): 562, 835, 1108, 1381, 1654, 1927, 2200, *2473, 2746, 3019, 3292, 3565, 3838, 3855, 158050, 232579, 307108, 377285, 447190, 451814, 460807, 521719, 530712, 535336, 605241, 679770, 833468, 907997, 977902, 982526, 987167, 

(17, 18): 631, 938, 1245, 1552, 1859, 2166, 2473, 2780, 3087, 3394, 3701, 4008, 4315, 4622, 4640, 199225, 293474, 387723, 476770, 571019, 659760, 675996, 764737, 858986, 1147258, 1241507, 1330248, 1341282, 

(18, 19): 704, 1047, 1390, 1733, 2076, 2419, 2762, 3105, *3448, 3791, 4134, 4477, 4820, 5163, 5506, 5525, 247970, 365619, 483268, 712410, 823561, 842732, 941210, 953883, 1071532, 1189181, 1430995, 1548644, 1666293, 1777444, 

(19, 20): 781, 1162, 1543, 1924, 2305, 2686, 3067, 3448, 3829, 4210, 4591, 4972, 5353, 5734, 6115, 6496, 6516, 305161, 450322, 595483, 740644, 878585, 1016146, 1023746, 1161307, 1176147, 1183747, 1321308, 1466469, 1611630, 1909571, 2054732, 2192293, 2199893, 2337454, 2352294, 

(20, 21): 862, 1283, 1704, 2125, 2546, 2967, 3388, 3809, 4230, *4651, 5072, 5493, 5914, 6335, 6756, 7177, 7598, 7619, 371722, 548963, 726204, 903445, 1072286, 1249527, 1417948, 1444009, 1595189, 1612430, 1789671, 1966912, 2330234, 2507475, 2684716, 2853137, 3030378, 3047619, 

(21, 22): 947, 1410, 1873, 2336, 2799, 3262, 3725, 4188, 4651, 5114, 5577, 6040, 6503, 6966, 7429, 7892, 8355, 8818, 8840, 448625, 662994, 877363, 1091732, 1510768, 1714973, 1929342, 1949230, 2163599, 2377968, 2592337, 3031260, 3245629, 3459998, 3664203, 3878572, 3898460, 

(22, 23): 1036, 1543, 2050, 2557, 3064, 3571, 4078, 4585, 5092, 5599, *6106, 6613, 7120, 7627, 8134, 8641, 9148, 9655, 10162, 10185, 536890, 793939, 1050988, 1308037, 1565086, 1811003, 2056414, 2068052, 2313463, 2347894, 2570512, 2593305, 2850354, 3107403, 3364452, 3633161, 3890210, 4147259, 4392670, 4404308, 4649719, 4906768, 4929561, 

(23, 24): 1129, 1682, 2235, 2788, 3341, 3894, 4447, 5000, 5553, 6106, 6659, 7212, 7765, 8318, 8871, 9424, 9977, 10530, 11083, 11636, 11660, 637585, 943394, 1249203, 1555012, 1860821, 2153934, 2459743, 2752304, 3058113, 3084081, 3389890, 3695699, 4001508, 4626397, 4932206, 5238015, 5530576, 5836385, 6142194, 6168162, 

(24, 25): 1226, 1827, 2428, 3029, 3630, 4231, 4832, 5433, 6034, 6635, 7236, *7837, 8438, 9039, 9640, 10241, 10842, 11443, 12044, 12645, 13246, 13271, 751826, 1113027, 1474228, 1835429, 2196630, 2904632, 3250833, 3612034, 3973235, 4002660, 4363861, 4725062, 5086263, 5462488, 5823689, 6184890, 6546091, 6892292, 7253493, 7614694, 7644119, 

(25, 26): 1327, 1978, 2629, 3280, 3931, 4582, 5233, 5884, 6535, 7186, 7837, 8488, 9139, 9790, 10441, 11092, 11743, 12394, 13045, 13696, 14347, 14998, 15024, 880777, 1304578, 1728379, 2152180, 2575981, 2999782, 3407333, 3814234, 3831134, 4238035, 4661836, 4695012, 5118813, 5542614, 5966415, 6830942, 7254743, 7678544, 8085445, 8102345, 8509246, 8933047, 9356848, 9390024, 

(26, 27): 1432, 2135, 2838, 3541, 4244, 4947, 5650, 6353, 7056, 7759, 8462, 9165, *9868, 10571, 11274, 11977, 12680, 13383, 14086, 14789, 15492, 16195, 16898, 16925, 1025650, 1519859, 2014068, 2508277, 3002486, 3496695, 3972652, 4466861, 4942116, 5436325, 5930534, 5967767, 6461976, 6956185, 7450394, 7963583, 8457792, 8952001, 9446210, 9921465, 

(27, 28): 1541, 2298, 3055, 3812, 4569, 5326, 6083, 6840, 7597, 8354, 9111, 9868, 10625, 11382, 12139, 12896, 13653, 14410, 15167, 15924, 16681, 17438, 18195, 18952, 18980, 1187705, 1760754, 2333803, 2906852, 3479901, 4052950, 5178636, 5730517, 6303566, 6876615, 6918223, 7491272, 8064321, 8637370, 9804663, 

(28, 29): 1654, 2467, 3280, 4093, 4906, 5719, 6532, 7345, 8158, 8971, 9784, 10597, 11410, *12223, 13036, 13849, 14662, 15475, 16288, 17101, 17914, 18727, 19540, 20353, 21166, 21195, 1368250, 2029219, 2690188, 3351157, 4012126, 4673095, 5334064, 5972297, 6609718, 6633266, 7270687, 7931656, 8592625, 8638938, 9299907, 9960876, 

(29, 30): 1771, 2642, 3513, 4384, 5255, 6126, 6997, 7868, 8739, 9610, 10481, 11352, 12223, 13094, 13965, 14836, 15707, 16578, 17449, 18320, 19191, 20062, 20933, 21804, 22675, 23546, 23576, 1568641, 2327282, 3085923, 3844564, 4603205, 5361846, 6120487, 6853898, 7612539, 8345080, 9103721, 9862362, 9913722, 

(30, 31): 1892, 2823, 3754, 4685, 5616, 6547, 7478, 8409, 9340, 10271, 11202, 12133, 13064, 13995, *14926, 15857, 16788, 17719, 18650, 19581, 20512, 21443, 22374, 23305, 24236, 25167, 26098, 26129, 1790282, 2657043, 3523804, 4390565, 5257326, 6124087, 6990848, 8696470, 9534401, 

(31, 32): 2017, 3010, 4003, 4996, 5989, 6982, 7975, 8968, 9961, 10954, 11947, 12940, 13933, 14926, 15919, 16912, 17905, 18898, 19891, 20884, 21877, 22870, 23863, 24856, 25849, 26842, 27835, 28828, 28860, 2034625, 3020674, 4006723, 4992772, 5978821, 6964870, 7950919, 8936968, 9892265, 

(32, 33): 2146, 3203, 4260, 5317, 6374, 7431, 8488, 9545, 10602, 11659, 12716, 13773, 14830, 15887, 16944, *18001, 19058, 20115, 21172, 22229, 23286, 24343, 25400, 26457, 27514, 28571, 29628, 30685, 31742, 31775, 2303170, 3420419, 4537668, 5654917, 6772166, 7889415, 9006664, 

Right at the very end of the above list, under number bases \(b=32\) and \(b=33\), we find \(26457\).$$ \begin{align} 26457_{_{10}}&=\text{ pqp }_{_{32}}\\&=\text{ o9o }_{_{33}} \end{align}$$What this means is that with \(o=24, p=25, q=26\) we have:$$\begin{align} 26457 &=25 \times 32^2+26 \times 32+25\\ &=24 \times 33^2+9 \times 33+24 \end{align}$$These numbers in fact form OEIS A279092:

A279092

Numbers that are nontrivially palindromic in two or more consecutive integer bases.

The initial members are:

10, 46, 67, 92, 98, 104, 121, 130, 135, 154, 178, 185, 191, 227, 232, 235, 277, 282, 292, 300, 326, 343, 373, 379, 410, 436, 446, 454, 455, 464, 483, 497, 543, 555, 562, 565, 631, 640, 646, 647, 651, 656, 676, 704, 738, 745, 781, 787, 797, 809, 835, 858, 862

Notice that in the StackExchange list for \(b=32\) and \(b=33\), \(18001\) is marked with an asterisk and so \(b=34\) is included as well:$$\begin{align} 18001_{_{10}}&=\text{ hih }_{_{32}}\\&=\text{ ghg }_{_{33}} \\&=\text{ fjf }_{_{34}} \end{align}$$Extracting the numbers with asterisks in the above list (remember the ∗ indicates that the number is also palindromic in a third consecutive base), we get:

(06, 07): 178 --> 454 b=6, 343 b=7, 262 b=8

(07, 08): 300 --> 606 b=7, 454 b=8, 363 b=9

(08, 09): 373 --> 565 b=8, 454 b=9, 373 b=10

(09, 10): 3360633 --> 6281826 b=9, 3360633 b=10 1995991 b=11

(10, 11): 676 --> 676 b=10, 565 b=11, 484 b=12

(12, 13): 1111 --> 787 b=12, 676 b=13, 595 b=14

(14, 15): 1702 --> 898 b=14, 787 b=15, 6a6 b=16

(16, 17): 2473 --> 9a9 b=16, 898 b=17, 7b7 b=18

(18, 19): 3448 --> aba b=18, 9a9 b=19, 8c8 b=20

(20, 21): 4651 --> bcb b=20, aba b=21, 9d9 b=22

(22, 23): 6106 --> cdc b=22, bcb b=23, aea b=24

(24, 25): 7837 --> ded base=24, cdc b=25, bfb b=26

(26, 27): 9868 --> efe b=26, ded b=27, cgc b=28

(28, 29): 12223 --> fgf b=28, efe b=29, dhd b=30

(30, 31): 14926 --> ghg b=30, fgf b=31, eie b=32

(32, 33): 18001 --> hih b=32, ghg b=33, fjf b=34

These numbers make up OEIS A279093:

A279093

Numbers that are nontrivially palindromic in three or more consecutive integer bases.

The initial members are:

178, 300, 373, 676, 1111, 1702, 2473, 3448, 4651, 6106, 7837, 9868, 12223, 14926, 18001, 21472, 25363, 29698, 34501, 39796, 45607, 51958, 58873, 66376, 74491, 83242, 92653, 102748, 113551, 125086, 137377, 150448, 164323, 179026, 194581, 211012, 228343, 246598

Here is a list of the first thousand such numbers. 

The comments to the OEIS entry include the statement that no numbers have been found that are palindromes in four successive bases. 130 is given as an example of a number that is palindromic in seven integer bases: $$11211_3 = 2002_4 = 202_8 = {\large aa}_{12} = 55_{25} = 22_{64} = 11_{129}$$but these bases do not include three consecutive integers, so 130 is not in the sequence. Incidentally, I'm creating this palindromic post on the 9th September 2021 and on the 12th September, the date can be written palindromically as 12/9/21.

Dali and the Unfolded Hypercube

Figure 1

In April of 1984, as a birthday present from my father, I acquired a TR-80 computer from Tandy with 32Kb of RAM and a keyboard but no disk drive. Data input and output were via a cassette tape. Display was via a connected TV. Using this I began my BASIC programming and one of my first creations was a simulation of a projectile being launched to hit a target. The essential formulae for the projectile's motion are:$$x(t)=v_0 \cos(\theta) \cdot t \text{ and } y(t)=v_0 \sin(\theta) \cdot t - 0.5 g t^2$$where we have:

• \(x(t)\) is horizontal position of projectile after time \(t\) with \(x=0\) when \(t=0\)
• \(y(t)\) is vertical position of projectile after time \(t\) with \(y=0\) when \(t=0\)
• \(v_0\) is initial velocity at time \(t=0\)
• \(\theta\) is angle of projection
• \(g\) is deceleration due to gravity \(\approx \) -9.8 \(ms^{-2}\)

Figure 1 shows the situation with an angle of 60° and an initial velocity fo 6 \(ms^{-1}\). What has all this to do with unfolding hypercubes? Well, April of 1984 marked my return to Mathematics after a long absence. In August of that year, I began my training as a Mathematics teacher and one of the assigned projects was an assignment on some aspect of Mathematics. I chose Mathematics and Art, a choice which led to me to the unfolded hypercube. In the University library, I came across Salvator Dali's 1954 Crucifixion (Corpus Hypercubus). See Figure 2.

Figure 2: source

It is a large oil painting, with dimensions of 194.3 cm × 123.8 cm (76.5 in × 48.7 in). Figure 3 shows the net of a hypercube (or tesseract).

Figure 3: source

What reminded me of my youthful investigation into the realm of the hypercube was a Mathologer video that I watched today titled The Iron Man hyperspace formula really works.

Figure 4 shows a screenshot that demonstrates information about the number of vertices, edges and faces in an \(n\)-dimensional cube where \(n\) varies from 0 to 4. Looking at it, we see that a zero dimensional "cube" consists of only a single point or vertex. The number of vertices is shown by the coefficients of the \(x^0\) terms, coloured red. A one dimensional "cube", corresponding to a line, has two vertices and one edge. The number of edges is shown by the coefficients of the \(x^1\) terms, coloured green. A two dimensional "cube", corresponding to a square, has four vertices, 4 edges and one face. The number of faces is shown by the coefficients of the \(x^2\) terms. The familiar three dimensional cube has 8 vertices, 12 edges and 6 faces, plus one cube. The number of cubes is shown by the coefficients of the \(x^3\) terms. The four dimensional hypercube has 16 vertices, 32 edges, 24 faces, 8 cubes and one hypercube. The vertices, edges, faces and cubes can be counted using the net shown in Figure 2.

Figure 4: source

Figure 5 shows another screenshot where a more general version of Euler's polyhedron formula is displayed. It begins with one dimension where a line of finite length does indeed have two end points or vertices. Thus V=2. For two dimensions, the number of vertices equals the number of edges e.g. consider a square. Thus V=E. For three dimensions, we have the familiar V-E+F=2 and for the hypercube we have V-E+F-C=0 because as can be seen in Figure 4, V=16, E=32, F=24 and C=8.

Figure 5: source

However, getting back to Dali's hypercube, I just finished watching an interesting talk about Dali from an American mathematician who met the artist on a number of occasions. The video is titled Math Encounters -- Encountering Salvador Dali in the Fourth Dimension. 

The video was uploaded in March of 2014 and the mathematician, Tom Banchoff, is still alive and now aged 83. He born on April 7th 1938. He has written a book, titled Beyond the Third Dimension Geometry, Computer Graphics, and Higher Dimensions, a digital copy of which I've managed to acquire. It was first published in 1990. The cover is shown in Figure 6.

Figure 6

The author makes mention of his book in this talk and mentions also Flatland which is a classic that I remember reading many years ago but exactly when I can't recall. Here is a little more about this 1884 book:

This masterpiece of science (and mathematical) fiction is a delightfully unique and highly entertaining satire that has charmed readers for more than 100 years. The work of English clergyman, educator and Shakespearean scholar Edwin A. Abbott (1838-1926), it describes the journeys of A. Square, a mathematician and resident of the two-dimensional Flatland, where women-thin, straight lines-are the lowliest of shapes, and where men may have any number of sides, depending on their social status. 

Through strange occurrences that bring him into contact with a host of geometric forms, Square has adventures in Spaceland (three dimensions), Lineland (one dimension) and Pointland (no dimensions) and ultimately entertains thoughts of visiting a land of four dimensions—a revolutionary idea for which he is returned to his two-dimensional world. Charmingly illustrated by the author, Flatland is not only fascinating reading, it is still a first-rate fictional introduction to the concept of the multiple dimensions of space. "Instructive, entertaining, and stimulating to the imagination." — Mathematics Teacher.

Figure 7 shows the frontispiece to the original publication. Dali was probably familiar with this book as he was very interested in such dimensional adventures. 

Figure 7

Speaking of Flatland's two dimensions, what would a hypercube look like in Flatland? One way it could appear is shown in Figure 8. I got this image from an interesting web page about hypercubes.

Figure 8: source

Figure 9 gives a clearer idea of why this is so. The shadow of the tesseract, itself the 3D shadow of the hypercube, is shown falling on a two dimensional surface.

Figure 9: source

To end this post, I'll return to Mathologer and his YouTube channel. In July of 2017, he posted an interesting video about hypercube shadows that's well worth a look.