Email: Jean-Pierre Luminet

Wednesday, July 17, 2013

TO: Bruce Camber
FM: Prof. Dr. Jean-Pierre Luminet
        Directeur de recherches au CNRS
        Laboratoire Univers et Théories (LUTH)
        Observatoire de Paris, 92195 Meudon

RE: Notations from the Planck Length to the Observable Universe

Hi Bruce:

I tried to understand the discrepancy between my calculation and that of Joe Kolecki. The reason is simple. Joe took as a maximum length in the universe the so-called Hubble radius, whereas in cosmology the pertinent distance is the diameter of the observable universe (delimited by the particle horizon), now estimated to be 93 billion light years, namely 8.8 10²⁶ meters. In my first calculation giving the result 206, I took the approximate 10²⁷ meters, and for the Planck length 10^-35 meters instead of the exact 1.62 10^-35 meters. Thus the right calculation gives 8.8 10²⁶m / 1.62 10^-35m = 5.5 10⁶¹ = 2^205.1. Thus the number of steps is 205 instead of 206. You can quote my calculation in your website.

– Jean-Pierre Luminet
http://luth.obspm.fr/~luminet/
http://luth.obspm.fr/~luthier/luminet/
http://luth.obspm.fr/~luthier/luminet/eluminet.html

Just the numbers…

These pages extend several overviews of the Big Board-little universe: (1) a Wiki-like working-draft, (2) one that is perhaps a little more speculative, and the most recent.

The first draft of Just the numbers was written in March 2012. There have been small edits as recently as May 2013.

What is simple? A point? What do we know about a point? There are many concepts to learn in just that one simple word. And then, how about two points? …three points? A rather idiosyncratic access path to the simple mathematics of points, lines, triangles and objects is to begin at the Planck length and simply multiply it by two. This type of exponential growth is called base-2 exponential notation. It creates a scale or orders of magnitude, a doubling of points (or layers, notations, steps, and/or vertices).

Powers-of-two and Exponentiation based on the Planck length. Herein it is referred to as Base-2 Exponential Notation (B2EN). Can the universe, from the smallest to the largest, be seen in a more meaningful way than by using base-ten scientific notation? B2EN renders greater granularity and a necessary relationality through nested geometries. The project originated with a series of five high-school geometry classes in December 2011. In looking at the five platonic solids, particularly the tetrahedron, the question was asked, “How far within could one go before hitting the wall of measurement, then hitting the Planck Length? Within a tetrahedron are four tetrahedrons and an octahedron. Within the octahedron are six octahedrons and eight tetrahedrons. Within each object, the edges can be divided in half and a new set of smaller objects observed. Yet, unlike the limitless paradox introduced of Zeno, (ca. 490 BC – ca. 430 BC), we know that the smallest possible measurement within space and time was defined mathematically back in and around 1899 by Max Planck. Though the Planck Constant, and Planck Length particularly, have not been universally accepted within the scientific community, it is a powerful concept based upon some of the most basic fundamentals of physics.

The Planck length is so small, it is written using exponential notation.

The number is 1.616199(97)x10^-35 meters. As a starting point we looked at many of the online references to the Planck length. In March 2012, there were just 276 Google links to that number (virtually none). Over the next few years, we suspect those references will grow substantially. Our guess is that it is one of the more important numbers within space and time.

Professor Laurence Eaves of the University of Nottingham in England has a delightful YouTube video that explains this length that is used to define a point (or a vertex).

In this simple exercise, take the Planck length and multiply it by 2, until we reach something that is measurable today (the diameter of a proton) and then to objects within the human scale, and finally to the edges of the observable universe. In one set of calculations, it only required 202+ notations or doublings. We have received excellent guesses based on excellent calculations from 202.34 to 206. In five columns, the first column is a Planck number based on the number of times the Planck length has been multiplied. The second column is the base-ten notations. The third column is the number of points (or vertices), the powers of two. The fourth column is for the incremental increase in size or length. And, the fifth column will be used for simple reflections.

B2EN	B10SN	Points, vertices or doublings	Length (meters)	Discussions, Examples,Information, Speculations:
0	1	2^0=1	1.616199(97)×10-35m	At the Planck Length, though it is a truly just a concept, let us take it as a given and that it is a special kind of point or singularity or vertex.
1	1	2^1=2	3.23239994×10-35m	At the first notation, there are two points (or vertices). Perhaps it is the shortest possible line, possibly the beginnings of a string and could open a discussion about some of the most basic questions in science.
2	1	2^2=4	6.46479988×10-35m	At the second notation there are four points (or vertices). There are several logical possibilities: (1) four vertices form a line, (2) four vertices form a jagged line of which various skewed triangles and polygons could be formed, (3) three vertices form a triangle that define a plane with the fourth vertex forming an imperfect or perfect tetrahedron that defines the first three dimensions of space.
3	2	2^3=8	1.292959976×10-34m	At the third doubling there are eight points (or vertices). The logical possibilities are now expanded to include placing the vertices either inside the tetrahedron, on the edges of the tetrahedron or outside the tetrahedron. Here multiplying could also involve dividing any of the six edges of the tetrahedron. If the vertices are are equally distributed on the edges of tetrahedron, an octahedron and four tetrahedrons begin to emerge (picture to be added). If added within (see the close-packing of equal spheres) tetrahedral close-packed structures emerge. If added externally, with just three additional vertices, a tetrahedral pentagon is created of five tetrahedrons (picture to be added). With all eight additional vertices added externally,a cube or hexahedron could be created.
4	2	2^4=16	2.585919952×10-34m	At the fourth doubling there are sixteen vertices. If any one of the vertices were to become a center vertex, and 10 vertices are extended from it, a tetrahedral icosahedron emerges (picture to be added). With twenty vertices a simple dodecahedron is possible. And with the icosahedron, all five platonic solids are accounted. Among the many possibilities, in another configuration, a cluster of four polytetrahedral clusters (a total of 20 tetrahedrons) begin to emerge and completes with twenty vertices (picture to be added). These vertices could also divide the edges of the internal four tetrahedrons and one octahedron. If the focus was entirely within the octahedron, the first shared center vertex of the octahedron would begin to be defined and by the 18th vertex the fourteen internal parts, eight tetrahedrons (one in each face) and the six octahedrons (one in each corner) would be defined (picture to be added).
5	2	2^5=32	5.171839904×10-34m	At the fifth notation, there are 32 vertices. Here there is a possibility for a cluster of eight tetrahedral pentagons to emerge and complete with 34 vertices.
6	3	2^6=64	1.0343679808×10-33m	At the sixth notation, there are 64 vertices. With just 43 of those vertices a hexacontagon could be created. It has 12 polytetrahedral clusters with an icosahedron in the middle.
7	3	2^7=128	2.0687359616×10-33m	By the seventh doubling, the possibilities become more textured. The results are not. Simple exponential notation based on the power of two is well documented. Of course, by using base-2 exponential notation and starting at the Planck length, necessary relations might be intuited.
8	3	2^8=256	4.1374719232×10-33m	Geometric complexification will be discussed. The nature of the perfect fittings, octahedrons and tetrahedrons, and the imperfect fitting, tetrahedrons making a pentastar or icosahedron, need review.
9	3	2^9=512	8.2749438464×10-33m	In that pentastar there is a 7.368 degree spread between tetrahedrons. That do not fit perfectly. That measurement is also 1.54 steradians and the effect within the icosahedron increases.
10	4	1024	1.65498876928×10-32m	At the tenth doubling, we look within an optimized Pentakis Dodecahedron where there is an outer layer of sixty tetrahedrons (twelve imperfect pentastars), an inner layer of irregular tetrahedrons, and an icosahedron of 20 tetrahedrons in the center. All are imperfect. Yet, within each tetrahedron there is a little perfection. There are four half-sized tetrahedrons and an octahedron all seated perfectly together. Within that octahedron, there are six octahedra and eight tetrahedra .
11	4	2048	3.30997752836×10-32m	_
12	4	4096	6.61995505672×10-32m	_
13	5	8192	1.323991011344×10-31m	_
14	5	16,384	2.647982022688×10-31m	_
15	5	32,768	5.295964045376×10-31m	_
16	6	65,536	1.0591928090752×10-30m	_
17	6	131,072	2.1183856181504×10-30m	_
18	6	262,144	4.2367712363008×10-30m	_
19	6	524,288	8.4735424726016×10-30m	_
20	7	1,048,576	1.69470849452032×10-29m	_
21	7	2,097,152	3.38941698904064×10-29m	more information
22	7	4,194,304	6.77883397808128×10-29m	_
23	8	8,388,608	1.355766795616256×10-28m	_
24	8	16,777,216	2.711533591232512×10-28m	_
25	8	33,554,432	5.423067182465024×10-28m	_
26	9	67,108,864	1.0846134364930048×10-27m	_
27	9	134,217,728	2.1692268729860096×10-27m
28	9	268,435,456	4.3384537459720192×10-27m	_
29	9	536,870,912	8.6769074919440384×10-27m	_
30	10	1,073,741,824	1.73538149438880768×10-26m	_
31	10	2,147,483,648	3.47076299879961536×10-26m	_
32	10	4,294,967,296	6.94152599×10-26m	_
33	11	8,589,934,592	1.3883052×10-25m	_
34	11	1.7179869×1011	2.7766104×10-25m	Actual number: 17,179,869,184 vertices
35	11	3.4359738×1011	5.5532208×10-25m	34,359,738,368
36	12	6.8719476×1011	1.11064416×10-24m	68,719,476,736
37	12	1.3743895×1012	2.22128832×10-24m	137,438,953,472
38	12	2.7487790×1012	4.44257664×10-24m	274,877,906,944
39	12	5.4975581×1011	8.88515328×10-24m	549,755,813,888
40	13	1.0995116×1012	1.77703066×10-23m	1,099,511,627,776
41	13	2.1990232×1012	3.55406132×10-23m	2,199,023,255,552
42	13	4.3980465×1012	7.10812264×10-23m	4,398,046,511,104
43	14	8.7960930×1012	1.42162453×10-22m	8,796,093,022,208
44	14	1.7592186×1013	2.84324906×10-22m	17,592,186,044,416
45	14	3.5184372×1013	5.68649812×10-22m	35,184,372,088,832
46	15	7.0368744×1013	1.13729962×10-21m	70,368,744,177,664
47	15	1.4073748×1014	2.27459924×10-21m	140,737,488,355,328
48	15	2.8147497×1014	4.54919848×10-21m	281,474,976,710,656
49	15	5.6294995×1014	9.09839696×10-21m	562,949,953,421,312
50	16	1.12589988×1015	1.81967939×10-20m	1,125,899,906,842,624
51	16	2.25179981×1015	3.63935878×10-20m	2,251,799,813,685,248
52	16	4.50359962×1015	7.27871756×10-20m	4,503,599,627,370,496
53	17	9.00719925×1015	1.45574351×10-19m	9,007,199,254,740,992
54	17	1.80143985×1016	2.91148702×10-19m	18,014,398,509,481,984
55	17	3.60287970×1016	5.82297404×10-19m	36,028,797,018,963,968
56	18	7.205759840×1016	1.16459481×10-18m	72,057,594,037,927,936
57	18	1.44115188×1017	2.32918962×10-18m	144,115,188,075,855,872
58	18	2.88230376×10 17	4.65837924×10-18m	288,230,376,151,711,744
59	18	5.76460752×1017	9.31675848×10-18m	576,460,752,303,423,488
60	19	1.15292150×1018	1.86335169×10-17m	1,152,921,504,606,846,976
61	19	2.30584300×1018	3.72670339×10-17m	2,305,843,009,213,693,952
62	19	4.61168601×1018	7.45340678×10-17m	4,611,686,018,427,387,904 (Quarks)
63	20	9.22337203×1018	1.49068136×10-16m	9,223,372,036,854,775,808 (Quarks, Photons)
64	20	1.84467440×1019	2.98136272×10-16m	18,446,744,073,709,551,616 (Neutrinos, Quarks, Photons)
65	20	3.68934881×1019	5.96272544×10-16m	36,893,488,147,419,100,000
66	21	7.37869762×1019	1.19254509×10-15m	73,786,976,294,838,200,000 (Protons, Fermions)
67	21	1.47573952×1020	2.38509018×10-15m	147,573,952,589,676,000,000 (Neutrons)
68	21	2.95147905×1020	4.77018036×10-15m	295,147,905,179,352,000,000 (Helium)
69	21	5.90295810×1020	9.54036072×10-15m	590,295,810,358,705,000,000 (Electron)
70	22	1.18059162×1021	1.90807214×10-14m	1,180,591,620,717,410,000,000 (Aluminum)
71	22	2.36118324×1021	3.81614428×10-14m	2,361,183,241,434,820,000,000 (Gold)
72	22	4.72236648×1021	7.63228856×10-14m	4,722,366,482,869,640,000,000
73	23	9.44473296×1021	1.52645771×10-13m	9,444,732,965,739,290,000,000
74	23	1.88894659×1022	3.05291542×10-13m	18,889,465,931,478,500,000,000
75	23	3.77789318×1022	6.10583084×10-13m	37,778,931,862,957,100,000,000
76	24	7.55578637×1022	1.22116617×10-12m	75,557,863,725,914,300,000,000
77	24	1.51115727×1023	2.44233234×10-12m	151,115,727,451,828,000,000,000
78	24	3.02231454×1023	4.88466468×10-12m	302,231,454,903,657,000,000,000
79	24	6.04462909×1023	9.76932936×10-12m	604,462,909,807,314,000,000,000
80	25	1.20892581×1024	1.95386587×10-11m	1,208,925,819,614,620,000,000,000
81	25	2.41785163×1024	3.90773174×10-11m	2,417,851,639,229,250,000,000,000
82	25	4.83570327×1024	7.81546348×10-11m	4,835,703,278,458,510,000,000,000
_	_	_________________	______________________	________________________
83	26	9.67140655×1024	.156309264 nanometers	9,671,406,556,917,030,000,000,000
			or 1.56309264×10-10m
84	26	1.93428131×1025	.312618528 nanometers	19,342,813,113,834,000,000,000,000
85	26	3.86856262×1025	.625237056 nanometers	38,685,626,227,668,100,000,000,000
_		_________________	______________________	________________________
86	27	7.73712524×1025	1.25047411 nanometers or	77,371,252,455,336,200,000,000,000
			or 1.25047411×10-9m
87	27	1.54742504×1026	2.50094822 nanometers	154,742,504,910,672,000,000,000,000
88	27	3.09485009×1026	5.00189644 nanometers	309,485,009,821,345,000,000,000,000
_		_________________	______________________	________________________
89	28	6.18970019×1026	10.0037929 nanometers	618,970,019,642,690,000,000,000,000
			or 1.00037929×10-8m
90	28	1.23794003×1027	20.0075858 nanometers	1,237,940,039,285,380,000,000,000,000
91	28	2.47588007×1027	40.0151716 nanometers	2,475,880,078,570,760,000,000,000,000
92	28	4.95176015×1027	80.0303432 nanometers	4,951,760,157,141,520,000,000,000,000
_		_________________	______________________	________________________
93	29	9.90352031×1027	160.060686 nanometers	9,903,520,314,283,042,199,192,993,792
			or 1.60060686×10-7m
94	29	1.98070406×1028	320.121372 nanometers	19,807,040,628,566,000,000,000,000,000
95	29	3.96140812×1028	640.242744 nanometers	39,614,081,257,132,100,000,000,000,000
_		_________________	______________________	________________________
96	30	7.92281625×1028	1.28048549 microns	79,228,162,514,264,300,000,000,000,000
			or 1.28048549×10-6m
97	30	1.58456325×1029	2.56097098 microns	158,456,325,028,528,000,000,000,000,000
98	30	3.16912662×1029	5.12194196 microns	316,912,650,057,057,000,000,000,000,000
_	_	_________________	______________________	________________________
99	31	6.33825324×1029	10.2438839 microns	633,825,300,114,114,700,748,351,602,688
			or 1.02438839×10-5m
100	31	1.26765065×1030	20.4877678 microns	1,267,650,600,228,220,000,000,000,000,000
101	31	2.53530130×1030	40.9755356 microns	2,535,301,200,456,450,000,000,000,000,000
102	31	5.07060260×1030	81.9510712 microns	5,070,602,400,912,910,000,000,000,000,000
_	_	_________________	______________________	________________________
103	32	1.01412052×1031	.163902142 millimeters	10,141,204,801,825,800,000,000,000,000,000
			or 1.63902142×10-4m
104	32	2.02824104×1031	.327804284 millimeters	20,282,409,603,651,600,000,000,000,000,000
105	32	4.05648208×1031	.655608568 millimeters	40,564,819,207,303,300,000,000,000,000,000
_	_	_________________	______________________	________________________
106	33	8.11296416×1031	1.31121714 millimeters	81,129,638,414,606,600,000,000,000,000,000
			or 1.31121714×10-3m
107	33	1.62259276×1032	2.62243428 millimeters	162,259,276,829,213,000,000,000,000,000,000
108	33	3.24518553×1032	5.24486856 millimeters	324,518,553,658,426,000,000,000,000,000,000
_		_________________	______________________	________________________
109	34	6.49037107×1032	1.04897375 centimeters	649,037,107,316,853,000,000,000,000,000,000
			or 1.04897375×10-2m
110	34	1.29807421×1033	2.09794742 centimeters	1,298,074,214,633,700,000,000,000,000,000,000
111	34	2.59614842×1033	4.19589484 centimeters	2,596,148,429,267,410,000,000,000,000,000,000
112	34	5.19229685×1033	8.39178968 centimeters	5,192,296,858,534,820,000,000,000,000,000,000
___	___	_________________	______________________	________________________
113	35	1.03845937×1034	16.7835794 centimeters or	10,384,593,717,069,600,000,000,000,000,000,000
			1.67835794×10-1m
114	35	2.0769437×1034	33.5671588 centimeters	20,769,187,434,139,300,000,000,000,000,000,000
115	35	4.1538374×1034	67.1343176 centimeters	41,538,374,868,278,600,000,000,000,000,000,000
___	___	_________________	______________________	_____________________
116	36	8.3076749×1034	1.3426864 meters	83,076,749,736,557,200,000,000,000,000,000,000
			or 52.86 inches
117	36	1.66153499×1035	2.6853728 meters	166,153,499,473,114,000,000,000,000,000,000,000
118	36	3.32306998×1035	5.3707456 meters	332,306,998,946,228,000,000,000,000,000,000,000
119	37	6.64613997×1035	10.7414912 meters	664,613,997,892,457,000,000,000,000,000,000,000
120	37	1.32922799×1036	21.4829824 meters	1,329,227,995,784,910,000,000,000,000,000,000,000
121	37	2.65845599×1036	42.9659648 meters	2,658,455,991,569,830,000,000,000,000,000,000,000
122	37	5.31691198×1036	85.9319296 meters	5,316,911,983,139,660,000,000,000,000,000,000,000
123	38	1.06338239×1037	171.86386 meters	10,633,823,966,279,300,000,000,000,000,000,000,000
124	38	2.12676479×1037	343.72772 meters	21,267,647,932,558,600,000,000,000,000,000,000,000
125	38	4.25352958×1037	687.455439 meters	42,535,295,865,117,300,000,000,000,000,000,000,000
126	39	8.50705917×1037	1.37491087 kilometers	85,070,591,730,234,600,000,000,000,000,000,000,000
127	39	1.70141183×1038	2.74982174 kilometers	170,141,183,460,469,000,000,000,000,000,000,000,000
128	39	3.40282366×1038	5.49964348 kilometers	340,282,366,920,938,000,000,000,000,000,000,000,000
129	40	6.04462936×1038	10.999287 kilometers	680,564,733,841,876,000,000,000,000,000,000,000,000
130	40	1.36112946×1039	21.998574 kilometers	1,361,129,467,683,750,000,000,000,000,000,000,000,000
131	40	2.72225893×1039	43.997148 kilometers	2,722,258,935,367,500,000,000,000,000,000,000,000,000
132	40	5.44451787×1039	87.994296 kilometers	5,444,517,870,735,010,000,000,000,000,000,000,000,000
133	41	1.08890357×1040	175.988592 kilometers	10,889,035,741,470,000,000,000,000,000,000,000,000,000
134	41	2.17780714×1040	351.977184 kilometers	2,177,807,148,294,000,000,000,000,000,000,000,000,000
135	41	4.355614296×1040	703.954368 kilometers	43,556,142,965,880,100,000,000,000,000,000,000,000,000
136	42	8.711228593×1040	1407.90874 kilometers	87,112,285,931,760,200,000,000,000,000,000,000,000,000
137	42	1.742245718×1041	2815.81748 kilometers	174,224,571,863,520,000,000,000,000,000,000,000,000,000
138	42	3.484491437×1041	5631.63496 kilometers	348,449,143,727,040,000,000,000,000,000,000,000,000,000
139	43	6.18970044×1041	11,263.2699 kilometers	696,898,287,454,081,000,000,000,000,000,000,000,000,000
140	43	1.23794009×1042	22,526.5398 kilometers	1,393,796,574,908,160,000,000,000,000,000,000,000,000,000
141	43	2.47588018×1042	45 053.079 kilometers	2,787,593,149,816,320,000,000,000,000,000,000,000,000,000
142	43	4.95176036×1042	90 106.158 kilometers	5,575,186,299,632,650,000,000,000,000,000,000,000,000,000
143	44	1.11503726×1043	180,212.316 kilometers	11,150,372,599,265,300,000,000,000,000,000,000,000,000,000
144	44	2.23007451×1043	360,424.632 kilometers	22,300,745,198,530,600,000,000,000,000,000,000,000,000,000
145	44	4.46014903×1043	720,849.264 kilometers	44,601,490,397,061,200,000,000,000,000,000,000,000,000,000
146	45	8.9202980×1043	1,441,698.55 kilometers	89,202,980,794,122,400,000,000,000,000,000,000,000,000,000
147	45	1.78405961×1044	2,883,397.1 kilometers	178,405,961,588,244,000,000,000,000,000,000,000,000,000,000
148	45	3.56811923×1044	5,766,794.2 kilometers	3.56812E+44
149	46	7.13623846×1044	11,533,588.4 kilometers	713,623,846,352,979,940,529,142,984,724,747,568,191,373,312
150	46	1.42724769×1045	23,067,176.8 kilometers	1.42725E+45
151	46	2.85449538×1045	46,134,353.6 kilometers	2,854,495,385,411,910,000,000,000,000,000,000,000,000,000,000
152	46	5.70899077×1045	92,268,707.1 kilometers	5.70899E+45
153	47	1.14179815×1046	184,537,414 kilometers	1.1418E+46
154	47	2.28359638×1046	369,074,829 kilometers	2.2836E+46
155	47	4.56719261×1046	738,149,657 kilometers	4.56719E+46
156	48	9.13438523×1046	1.47629931×1012 meters	9.13439E+46
157	48	1.826877046×1047	2.95259863×1012 meters	1.82688E+47
158	48	3.653754093×1047	5.90519726×1012 meters	3.65375E+47
159	49	7.307508186×1047	1.18103945×1013 meters	7.30751E+47
159	49	7.307508186×1047	1.18103941 ×1013 meters	730,750,818,665,451,000,000,000,000,000,000,000,000,000,000,000
160	49	1.461501637×1048	2.36207882 ×1013m	1.4615E+48
161	49	2.923003274×1048	4.72415764 ×1013m	2.923E+48
162	49	5.846006549×1048	9.44831528 ×1013m	5.84601E+48
163	50	1.16920130×1049	1.88966306×1014m	1.1692E+49
164	50	2.33840261×1049	3.77932612×1014m	2.3384E+49
165	50	4.67680523×1049	7.55865224×1014m	4.67681E+49
166	51	9.35361047×1049	1.5117305×1015m	9.35361E+49
167	51	1.87072209×1050	3.0234609×1015m	1.87072E+50
168	51	3.74144419×1050	6.0469218×1015m	3.74144E+50
169	52	7.48288838×1050	1.20938436×1016m	7.48289E+50
170	52	1.49657767×1051	2.41876872×1016m	1.49658E+51
171	52	2.99315535×1051	4.83753744 ×1016m	2.99316E+51
172	52	5.98631070×1051	9.67507488 ×1016m	5.98631E+51
173	53	1.19726214×1052	1.93501504 ×1017m	1.19726E+52
174	53	2.39452428×1052	3.87002996 ×1017m	2.39452E+52
175	53	4.78904856×1052	7.74005992 ×1017m	4.78905E+52
176	54	9.57809713×1052	1.54801198×1018m	9.5781E+52
177	54	1.91561942×1053	3.09602396×1018m	1.91562E+53
178	54	3.83123885×1053	6.19204792×1018m	3.83124E+53
179	55	7.66247770×1053	1.23840958×1019m	7.66248E+53
180	55	1.53249554×1054	2.47681916×1019m	1.5325E+54
181	55	3.06499108×1054	4.95363832×1019m	3.06499E+54
182	55	6.12998216×1054	9.90727664×1019m	6.12998E+54
183	56	1.22599643×1055	1.981455338×1020m	1.226E+55
184	56	2.45199286×1055	3.96291068×1020m	2.45199E+55
185	56	4.90398573×1055	7.92582136×1020m	4.90399E+55
186	57	9.80797146×1055	1.58516432×1021m	9.80797E+55
187	57	1.96159429×1056	3.17032864×1021m	1.96159E+56
188	57	3.92318858×1056	6.34065727 ×1021m	3.92319E+56
189	58	7.84637716×1056	1.26813145 ×1022m	7.84638E+56
190	58	1.56927543×1057	2.53626284×1022m	1.56928E+57
191	58	3.13855086×1057	5.07252568×1022m	3.13855E+57
192	59	6.27710173×1057	1.01450514×1023m	6.2771E+57
193	59	1.25542034×1058	2.02901033×1023m	1.25542E+58
194	59	2.51084069×1058	4.05802056×1023m	2.51084E+58
195	59	5.02168138×1058	8.11604112×1023m	5.02168E+58
196	60	1.00433628×10	1.62320822×1024m	1.00434E+59
197	60	2.0086725×1059	3.24641644×1024m	2.00867E+59
198	60	4.01734511×1059	6.49283305×1024m	4.01735E+59
199	61	8.03469022×1059	1.29856658×1025m	8.03469E+59
200	61	1.60693804×1060	2.59713316×1025m	1.60694E+60
201	61	3.21387608×1060	5.19426632 ×1025m	3.21388E+60
		_________________	______________________	_____________________________________________
202	61	6.42775217×1060	1.03885326×1026 meters	6.42775E+60
203	62	1.28555043×1061	2.07770658×1026 meters	1.28555E+61
204	62	2.57110087×1061	4.15541315×1026 meters	2.5711E+61
205	62	5.14220174×1061	8.31082608×1026 meters	5.1422E+61
206	63	1.028440348×1062	1.662165216×1027 meters	1.0284E+62

Everything Starts Most Simply. Therefore, Might It Follow That The Planck Length Becomes The Next Big Thing?

Please note: This document is subject to updates by Bruce Camber and many others.

Propaedeutics: Analyze three very simple concepts taken from a high school geometry class, (1) the smallest-and-largest measurement of a length, (2) dividing and multiplying by 2, and (3) nested-embedded-and-meshed geometries. Though initially a simple thought exercise (hardly an experiment), our students quickly developed a larger vision to create a working framework to categorize and relate everything in the known universe. Though appearing quite naive and overly ambitious in its scope, the work began at the Planck Length and proceeded to the Observable Universe in somewhere over 201 base-2 exponential notations. That range of notations is examined and the unique place of the first sixty notations is reviewed. This simple mathematical progression and the related geometries, apparently heretofore not examined by the larger academic community, are the praxis; interpreting the meaning of it all is the theoria, and here we posit a very simple foundation to open those discussions. Along this path it seems we will learn how numbers are the function and geometries are the form, how each is the other’s Janus face, and perhaps even how time is derivative of number and space derivative of geometry.

Simple Embedded Geometries, The Initial Framework For A Question

Observing how the simplest geometric objects are readily embedded within each other, a high school geometry class¹ asked a similar question to that asked by Zeno (circa 430 BC) centuries earlier.² “How many steps inside can we go before we can go no further?” The students had learned about the Planck Length, a conceptual limit of 1.616199(97)x10^-35 meters. Using base-2 exponential notation, these students rather quickly discovered that it took just over 101 steps going within to get into the range of the Planck Length. For this exercise they followed just two geometrical objects, the simple tetrahedron and the octahedron. Within that tetrahedron is an octahedron perfectly enclosed within it. Also, within each corner are four half-sized tetrahedrons.

Illustration 1: The simple tetrahedron, the center triangle
being a face of an octahedron

We went inside again. At each notation or step we simply selected an object and divided the edges in half and connected the dots. Perfectly enclosed within the octahedron are six half-sized octahedrons in each of the six corners and eight half-sized tetrahedrons in each of the eight faces.

Illustration 2: An octahedron with its simplest internal parts.
Four pivotal hexagonal plates are outlined (red-white-blue-yellow); all surround the center point.

Selecting either a tetrahedron or octahedron, it would seem that one could divide-by-2 or multiply-by-2 each of the edges without limit. If we take the Planck Length as a given, it is not possible at the smallest scale. And, if we take the measurements of the Sloan Digital Sky Survey (SDSS III),  Baryon Oscillation Spectroscopic Survey (BOSS)³ as a given, there are also apparent limits within the large-scale universe — it is called the Observable Universe.

Also, observe how the total number of tetrahedrons and octahedrons increases at each doubling. At the next doubling there are a total of 10 octahedrons and 24 tetrahedrons. On the third doubling, there are 84 octahedrons and 176 tetrahedrons, and then on the fourth, 680 octahedrons and 1376 tetrahedrons. On the fifth step within, there are 10944 tetrahedrons and 5456 octahedrons. The numbers become astronomically large within 101 steps. It is more aggressive than the base-2 exponential notation used with the classic wheat and chessboard story⁴ which, of course, is only 64 steps or notations.

The following day we followed the simple math going out to the edges of the Observable Universe. There were somewhere between 101 to 105 steps (doublings or notations) to get out in the range of that exceeding large measurement, 1.03885326×10²⁶ meters. By combining these results, we had the entire “known” universe, from the smallest to the largest measurements in 202.34+ (calculation by NASA’s Joe Kolecki) to 205.11+ (calculation by Jean-Pierre Luminet) notations⁵. At the same time the growth of the number of objects by multiplying or dividing became such a large number that it challenged our imaginations. We had to learn to become comfortable with numbers in new ways — both exceedingly large and exceedingly small, and the huge numbers of objects.

Not long into this exploration it was realized that to achieve a consistent framework for measurements, this simple model for our universe ought to begin with the Planck Length (ℓP). It was a very straightforward project to multiply by 2 from the ℓP to the edges of the Observable Universe (OU). That model first became a rather long chart that was dubbed the Big Board – little universe.⁶ And then, sometime later we began converting it to a much smaller table⁷ (also, a working draft).

This simple construction raised questions about which we had no answers:
1. Planck Length. Why is the Planck Length the right place to start? Can it be multiplied by 2? What happens at each step?
2. The first 65 Notations. Although we initially started with a tetrahedron with edges of one meter, in just 50 notations, dividing by 2, we were in the range of the size of a proton.⁸ It would require another sixty-five steps within to get to the Planck Length. It begs the question, “What happens in each of those first 65 doublings from the Planck Length?”
3. Embedded Geometries. When we start at the human scale to go smaller by dividing by 2, the number of tetrahedrons and octahedrons at each notation are multiplied by 4 and 1 within the tetrahedron and by 8 and 6 within the octahedron. That results in an astronomical volume of tetrahedrons and octahedrons as we approach the size of a proton. What does it mean and what can we do with that information?

Starting at the Planck Length, a possible tetrahedron can manifest at the second doubling and an octahedron could manifest at the third doubling. Thereafter, growth is exponential, base-4 and base-1 within the tetrahedron and base-8 and base-6 within the octahedron. To begin to understand what these numbers, the simple math, and the geometry could possibly mean, we turned to the history of scholarship particularly focusing on the Planck Length.

Discussions about the meaning of the Planck Length.

Physics Today (Mead – Wilczek discussions).⁹ Though formulated in 1899 and 1900, the Planck Length received very little attention until C. Alden Mead in 1959 submitted a paper proposing that the Planck Length and Planck Time should “…play a more fundamental role in physics.” Though published in Physical Review in 1964, very little positive feedback was forthcoming. Frank Wilczek in that 2001 Physics Today article comments that “…C. Alden Mead’s discussion is the earliest that I am aware of.” He posited the Planck constants as real realities within experimental constructs whereby these constants became more than mathematical curiosities.

Frank Wilczek continued his analysis in several papers and books and he has personally encouraged the students and me to continue to focus on the Planck Length. We are.

The simple and the complex

A very simple logic suggests that things are always simple before they become complex. It seems that I adopted this idea while growing up as a child; my father would ask, “Is there an even more simple solution?” Complex solutions make us feel smarter and wiser, yet the opposite is most often true. When teaching students from ages 12 to 18, one must always start with the simplest new concepts and build on them slowly. Then, a good teacher might challenge the students to see something new, “If you can, find a more simple solution.”

Our class was basic science and mathematics, focusing on geometry. My assignment was to introduce the students to the five platonic solids. Yet, by our third time together, we were engaging the Planck Length. Is it a single point? Is it a vertex making the simplest space? What else could it be? Can it be more than just a physical measurement? Are we looking at point-free geometry? ¹⁰ Is this a pre-structure for group theory?¹¹ Speculations quickly got out of hand.

We knew we would be coming back to those questions over and over again, so we went on. We had to assume that the measurement could be multiplied by 2. We attributed that doubling to the thrust of life.¹² So, now we have two points, or two vertices, or a line, and a larger space of some kind. Prof. Dr. Freeman Dyson¹³ in a personal email suggests, “Since space has three dimensions, the number of points goes up by a factor eight, not two, when you double the scale.” We liked that idea; it would give us more breathing room. However, when we realized there would be an abundance of vertices, we decided to continue to multiply by two. We wanted to establish a simple platform using base-2 exponential notation especially because it seemed to mimic life’s cellular division and chemical bonding.

The first 60 doublings, layers, steps, or notations

Facts & Guesses. If taken-as-a-given, the Planck Length is a primary vertex and it can be multiplied by 2. The exponential progression of numbers becomes a simple fact. Guessing about the meaning of the progression is another thing. And to do so, we must hypothesize, possibly just hypostatize, the basic meanings and values. In our most far-reaching thoughts, this construct seems to open up possibilities to intuit an infrastructure or pre-structure that just might-could create a place for all that scholarship that doesn’t appear to have a grid and inherent matrix — philosophies, psychologies, thoughts and ideas throughout time. So herein we posit a simple fact and make our most speculative guesses:

• Within the first ten doublings, there are over 1000 vertices. Perhaps we might think about Plato’s Eidos, the Forms.
• Within twenty doublings, there are over a million vertices. What about Aristotle’s Ousia or Categories?
• Within thirty steps, there are over a billion vertices. Perhaps we could hypostatize Substances, a fundamental layer that anticipates the table of elements or periodic table.
• Within forty layers, there are over a trillion vertices. Might we intuit Qualities?
• Within 50 doublings, there over a quadrillion vertices. How about layers for Primary Relations, the precursors of subjects and objects?
• Between the 50th and 60th notation, still much smaller than the proton, there are over a quintillion vertices. Perhaps Systems and The Mind, and every possible manifestation of a mind, awaits its place within this ever-growing matrix or grid.
• The simple mathematics for these notations, virtually the entire small-scale universe, appears to be the domain of elementary cellular automata going back to the 1940s work of John von Neumann, Nicholas Metropolis and Stanislaw Ulam, and the more recent work of John Conway and his Game of Life, and most recently the work of Stephen Wolfram and his research behind A New Kind of Science.

With so many vertices, one could build a diversity of constructions, then ask the question, “What does it mean?” Our exercise with the simplest math and simple concepts is the praxis. We have begun to turn to the history of scholarship to begin to deem the theoria and begin to see if any of our intuitions might somehow fit.

We knew our efforts were naive, surely a bit idiosyncratic (as physicist, John Baez¹⁴ had characterized them), but we were attempting to create a path that would take us from the simplest to the most complex. If we stayed with our simple math and simple geometries, we figured that we did not have to understand the dynamics of protons, fermions, scalar constraints and modes, gravitational fields, and so so much more. That could come later.

Although not studied per se, these 60 notations have been characterized throughout the years. Within the scientific age, it has been discussed as the luminiferous aether (ether).¹⁵ Published in 1887 by Michelson–Morley, their work put this theory to rest for about a century. Yet, over the years, the theories around an aether have been often revisited. The ancient Greek philosophers called it quintessence¹⁵ and that term has been adopted by today’s theorists for a form of dark energy.

Theories abound.

Oxford physicist-philosopher Roger Penrose¹⁶ calls it, Conformal Cyclic Cosmology made popular within his book, Cycles of Time. Frank Wilczek simply calls this domain, the Grid,¹⁷ and the most complete review of it is within his book, The Lightness of Being.

We know with just two years of work on this so-called Big Board – little universe chart and much less time on our compact table, we will be exploring those 60-to-65 initial steps most closely for years to come. This project will be in an early-stage development for a lifetime.

From Parameters to Boundaries and Boundary Conditions

This construction with its simple nested geometries and simple calculations (multiplying the Planck Length by 2 as few as 202.34 times to as many as 205.11 times) puts the entire universe in an mathematically ordered set and a geometrically homogeneous group. Although functionally interesting, quite simple and rather novel, is it useful?

Some of the students thought it was. This author thought it was. And, a few scholars with whom we have spoken encouraged us. So the issue now is to continue to build on it until it has some real practical philosophical, mathematical, and scientific applicability.

Taking our three simple parameters just as they have been given, (1) the Planck Length, (2) multiplication by 2 and (3) Plato’s simplest geometry, what more can we say about this simple construct? Let me go out on a limb here:

1. Parameters. These parameters have functions; each creates a simple order and that order creates continuity. The form is order and the function is “to create continuity or its antithesis, discontinuity.” As a side note, one could observe, that this simple parameter set is also the beginning of memory and intelligence.

2. Relations. The parameters all work together to form a simple relation. From four points, a potential tetrahedron, simple symmetries are introduced. With eight points, the third doubling, a potential simple octahedron could become manifest. All the parameters work together to provide a foundation for additional simple functions to manifest. The form is the relation and the function is “to make and break symmetries.”

3. Dynamics. Our simple parameters, now manifesting real relations that have the potential to be extended in time, create a foundation for dynamics, all dynamics. That is the form with the potential to become a category, and the function is to create various harmonies or   to create disproportion, imbalance, or disagreement. Dynamics open us to explore such concepts as periodicity, waves, cycles, frequency, fluctuations, and more. And, this third parameter set, dynamics-harmony, necessarily introduces our perception of time. With this additional parameter set we begin to intuit what might give rise to the fullness of any moment in time and of time itself. Also, perspectivally, these parameter sets, on one side, just might could summarize perfection or a perfected moment in time, and on the other side, imperfection or quantum physics.

Please note that our use of the double modal, might could, is a projection for future, intense analysis and interpretation. It is a common expression in the New Orleans area.

Perfections and Imperfections.

The first imperfection can occur very early within the notations (doublings-steps-vertices). With the first doubling there are two vertices (the smallest line or smallest-possible string). At the next doubling, there are four vertices; a perfect tetrahedron could be rendered. It is the simplest three-dimensional form defined by the fewest number of vertices and equal angles. There are other logical possibilities: (1) four vertices form a longer line or string, (2) four vertices form a jagged line or string of which various skewed triangles and polygons could be formed, (3) three vertices form a triangle that defines a plane with the fourth vertex forming an imperfect tetrahedron that opens the first three dimensions of space. Five vertices can be used to create two tetrahedrons with a common face. Six vertices could be used to create an octahedron or three abutting tetrahedrons (two faces are shared).

The third doubling renders eight vertices. With just seven of those vertices, a pentagonal cluster of five tetrahedrons can be inscribed (Illustration 3), however, there is a gap of about 7.36° (7° 21′) or less than 1.5° between each face.¹⁹ There are many other configurations of a five-tetrahedral construction that can be created with those seven vertices. These will be addressed in a separate article. For our discussions here, it seems that each suggests a necessarily imperfect construction. The parts only fit together by stretching them out of their simple perfection. One might speculate that the spaces created within these imperfections could also provide room for movement or fluctuation.

Illustration 3: The earliest analysis of these five regular tetrahedra sharing one edge appears to be the work of F. C. Frank and J.S. Kaspers in their 1959 analysis of complex alloy structures. (See footnote 19 for more details on this reference).

With all eight vertices, a rather simple-but-complex figure can be readily constructed with six tetrahedrons, three on either side of a rather-stretched pyramid filling an empty space between each group. This figure has many different manifestations using just eight vertices. Between seven and eight vertices is a key step in this simple evolution. Both figures can morph and change in many different ways, breaking-and-making perfect and imperfect constructions.

A few final flights of imagination

In one’s most speculative, intuitive moments, one “might-could”  see these constructions as a way of engaging the current work with the Lie Group,²⁰ yet here may begin a different approach to continuous transformations groups. Just by replicating these eight vertices, a tetrahedral-octahedral-tetrahedral (TOT) chain emerges.²¹ Here octahedrons and two tetrahedrons are perfectly aligned and a simple structure reaching from the smallest to the largest readily emerges and tiles the universe. Then, there is yet another very special hexagonal tiling application to be studied within the octahedron by observing how each of the four hexagonal plates interact with all congruent tetrahedrons.

Within the all the following notations simplicity begets complexity. Structures become diverse. And, grids of potential and a matrix of possibilities are unlocked.

Footnotes: (Work-in-progress)
¹ Our Start Date:  Monday, December 19, 2011 Bruce Camber substituted for the geometry teacher within the John Curtis Christian High School, just up river from New Orleans. The concept of a Big Board – little universe developed within the context of these classes.

² Zeno’s paradoxes, Zeno of Elea (ca. 490 BC – ca. 430 BC), a pre-Socratic Greek philosopher and member of the Eleatic School founded by Parmenides known for his paradoxes to understand the finite and infinite

³ Base-2 Exponential notation: For most students, the wheat & chessboard example is their introduction to exponential notation. Wikipedia provides an overview.

⁴ “Most Precise Measurement of Scale of the Universe,” Jennifer Ouellette, Discover Magazine, April 6, 2012

⁵ Prof. Dr. Jean-Pierre Luminet, on Wednesday, July 17, 2013, wrote: “I tried to understand the discrepancy between my calculation and that of Joe Kolecki. The reason is simple. Joe took as a maximum length in the universe the so-called Hubble radius, whereas in cosmology the pertinent distance is the diameter of the observable universe (delimited by the particle horizon), now estimated to be 93 billion light years, namely 8.8 10^26 m. In my first calculation giving the result 206, I took the approximate 10^27 m, and for the Planck length 10^(-35) m instead of the exact 1.62 10^(-35) m. Thus the right calculation gives 8.8 10^26 m / 1.62 10^(-35) m = 5.5 10^(61) = 2^(205.1). Thus the number of steps is 205 instead of 206. You can quote my calculation in your website.” – Jean-Pierre Luminet, Directeur de recherches au CNRS, Laboratoire Univers et Théories (LUTH), Observatoire de Paris, 92195 Meudon Cedex http://luth.obspm.fr/~luminet/

⁶ Big Board – little universe, a five foot by one foot chart that begins with the Planck Length and uses exponential notation to go to the width of a human hair in 102 steps and to the edges of the observable universe in 202.34-to-205.11 notations, or steps, or doublings.

⁷ Universe Table, ten columns by eleven rows, this table is made to be displayed on Smartphones and every other form of a computer. At the time of this writing, Version 1.0.0.2. was posted..

⁸ Very large Numbers:  Taking just the octahedron, the calculation is: 6⁶⁵=3.8004172ex10⁵⁰ octahedrons and 8⁶⁵= 5.0216814e⁵⁸ tetrahedrons. Add to that, with the tetrahedrons at each step are four tetrahedrons: 4⁶⁵=1.3611295ex10³⁹ and the additional octahedron within it at each step: 1⁶⁵=65

⁹Frank Wilczek, the head of the Center for Theoretical Physics at MIT and a 2004 Nobel Laureate has a series of articles about the Planck Length within Physics Today. Called Scaling Mt. Planck, these are all well-worth the read. His book, The Lightness of Being, to date, is his most comprehensive summary.

¹⁰ Point-free geometry, a concept introduced by A. N. Whitehead in 1919/1920, was further refined in 1929 within his publication of the book, Process & Reality. More recent studies within mereotopology continue to extend Whitehead’s initial work.

¹¹Group Theory and Speculations: One might speculate that group theory, with its related subjects such as combinatorics, fields, representation theory, system theory and Lie transformation groups, all apply in some way to the transformation from one notation to the next. Yet, two transformations seem to beg for special attention. One is from the Human Scale to the Small Scale and the other from the Human Scale to the Large Scale. If there are 202.34-to-205.11 notations, our focus might turn to steps 67 to 69 at the small scale and 134 to 138 at the large scale universe. One’s speculations might could run ahead of one’s imaginative sensibilities. For example, at the transformation to the small scale, approximately in the range of the diameter of a proton, one could hypostatize that this is where the number of embedded geometries begins to contract to begin to approach the most-simple structure of the Planck Length. It would follow that within the small-scale all structures would necessarily be shared. Perhaps the proton is some kind of a boundary for individuation. That is, the closer one gets to the singularity of the Planck Length, the more those basic geometric structures within the notation are shared. Because this structure currently appears to be beyond the scope of measuring devices, we could refer to these notations as a hypostatic science, whereby hypotheses, though apparently impossible to test, are still not beyond the scope of imagination. Also, as the large scale is approached, somewhere between notations 134 to 138, there might be a concrescence that opens the way to even more speculative thinking. Though not very large — between 248 miles (notation 134) and- 3500 miles (notation 138) — it might appear to be silly, truly nonsensical, to begin the search for the Einstein-Rosen bridges or wormholes! That’s certainly science fiction. Yet, if we let an idea simmer for awhile, maybe workable insights might-could begin to emerge.

¹² Purdue & NSF:  Although the term, Thrust of Life, is used within religious and philosophical studies, it is also the subject of continuous scientific study by groups such as the Center for Science for of Information (Purdue University) through funding from the National Science Foundation.

¹³ Freeman Dyson:  Personal email to me regarding multiplying the Planck Length by 2, he said: “Since space has three dimensions, the number of points goes up by a factor eight, not two, when you double the scale.” Certainly a cogent comment, however, given we have seemingly more than enough vertices, we decided on the first pass to continue to multiply by 2 to create an initial framework from which attempt to grasp what was important and functional.

¹⁴ Anonymous:  Personal email to me regarding the initial posting for Wikipedia, he said: “…it’s certainly an idiosyncratic view, not material for an encyclopedia.”  To which we say, “Thank you.”

¹⁵ The luminiferous aether was posited by many of the leading scientists of the 18th century, Sir Issac Newton (Optiks) being the most luminous. The Michael-Morley experiments of 1887 put the theory on hold such that the theory of relativity and quantum theory emerged. Yet, research to understand this abiding concept has not stopped. And, it appears that the editorial groups within Wikipedia are committed to updating that research.

¹⁶ Quintessence, the Fifth Element in Plato’s Timaeus, has been used interchangeably with the aether aether. It has a long philosophical history. That the word has been adopted in today’s discussion as one of the forms of dark energy tells us how important these physicists believe dark energy is.

¹⁷ Roger Penrose inspired the 1998 book, The Geometric Universe: Science, Geometry, and the Work of Roger Penrose. Surely Penrose is one of the world’s leading thinkers in mathematics and physics. He has been in the forefront of current research and theory since 1967, however, his work on Conformal Cyclic Cosmology is not based on simple mathematics or simple geometries. It is based on the historic and ongoing tensions within his disciplines. Though his book, Cycles of Time, written for the general population, it is brings all that history and tension with it.

¹⁸ Frank Wilczek has written extensively about the Planck length. He recognizes its signature importance within physics. When we approached him with our naive questions via email in December 2012, we did not expect an answer, but, we received one. It was tight, to the point, and challenged us to be more clear. Given he was such a world-renown expert on such matters, we were overjoyed to respond. The entire dialogue will go online at some time. He is a gracious, thoughtful thinker who does not suffer fools gladly. And because we believe, like he does, in beauty and simplicity, perhaps there will be a future dialogue that will further embolden us.

¹⁹ Frank, F. C.; Kasper, J. S. (1958), “Complex alloy structures regarded as sphere packings. I. Definitions and basic principles”, Acta Crystall. 11. and Frank, F. C.; Kasper, J. S. (1959), and “Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures”, Acta Crystall. 12. More recently, this construct has been analyzed by the following:
(1) “A model metal potential exhibiting polytetrahedral clusters” by Jonathan P. K. Doye, University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom, J. Chem. Phys. 119, 1136 (2003) The compete article is also available at ArXiv.org as a PDF: http://arxiv.org/pdf/cond-mat/0301374‎
(2) “Polyclusters” by the India Institute of Science in Bangalore has many helpful illustrations and explanations of crystal structure. PDF: http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf
(3) “Mysteries in Packing Regular Tetrahedra” Jeffrey C. Lagarias and Chuanming Zong, a focused look at the history. To download: http://www.ams.org/notices/201211/rtx121101540p.pdf

²⁰ The work of Sophus Lie (1842 – 1899), a Norwegian mathematician, not only opened the way to the theory of continuous transformation groups for all of mathematics, it has given us a pivot point within group theory by which to move our analysis from parameters to boundary conditions and on to transformations between each notation. We are hoping that we are diligent enough to become Sophus Lie scholars.

²¹ “New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra” John H. Conway, Yang Jiao, and Salvatore Torquato. To download: http://cherrypit.princeton.edu/papers/paper-312.pdf

About the author(s)
In 1970 Bruce Camber began his initial studies of the 1935 Einstein-Podolsky-Rosen (EPR) thought experiment. In 1972 he was recruited by the Boston University School of Theology based on (1) his research of perfected states in space-time through work within a think tank in Cambridge, Massachusetts, (2) his work within the Boston University Department of Physics, Boston Colloquium for the Philosophy of Science, and (3) his work with Arthur Loeb (Harvard) and the Philomorphs. With introductions by Victor Weisskopf (MIT) and Lew Kowarski (BU), he went to CERN on two occasions, primarily to discuss the EPR paradox with John Bell. In 1979, he coordinated a project at MIT with the World Council of Churches to explore shared first principles between the major academic disciplines represented by 77 peer-selected, leading-living scholars. In 1980 he spent a semester with Olivier Costa de Beauregard and Jean-Pierre Vigier at the Institut Henri Poincaré focusing on the EPR tests of Alain Aspect at the Orsay-based Institut d’Optique. In 1994, following the death of another mentor, David Bohm, Camber re-engaged simple interior geometries based on several discussions with Bohm and his book, Fragmentation & Wholeness. In 1997 he made the molds to manufacture clear plastic models of the tetrahedron and octahedron. These models are used just above. In 2002, he spent a day with John Conway at Princeton to discuss the simplicity of the interior parts of the tetrahedron and octahedron. In 2011, he challenged a high school geometry class to use base-2 exponential notation to follow the interior structure of basic geometries from the Planck Length and to the edges of the Observable Universe.    The first iteration of the Big Board-little universe was created. In September 2013, the first iteration of the Universe Table was created. In 2014, a chart to enclose a moment of perfection extending from the chaotic to the good was introduced.

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© 2014, Small Business School, Bruce Camber