Planck Time to the Age of The Universe alongside Planck Length to The Observable Universe

Early in December 2014 we started this page to follow-up that earlier work on just the Planck Length. We began that effort three years earlier (December 2011) in our local high school’s geometry classes. Because we will continue to find obvious errors (from simple mathematics to our interpretation) of the chart below, this page will be subject to frequent updates.

Background: We had been asking around the scholarly community, “Has anyone done a progression of the Planck Time to the Age of the Universe using base-2 exponential notation (a fancy way of saying, multiplying by 2)?” We did it from the Planck Length to the Observable Universe and had wanted to compare that progression to Planck Time.

Going from the smallest to the largest is a simple ordering logic. Using Max Planck’s smallest possible measurements to go to the known limits seems like an exercise high school students should do.

Here we introduce the simple math from the Planck Time to the Age of the Universe.

In July 2014, Prof. Dr. Gerard ‘t Hooft and Stefan Vandoren published a very helpful book, Time in Powers of Ten, a base-10 chart. We were looking for a base-2 chart which would be 3.333+ times more granular. We could not find it anywhere so this page is our working draft, our starting point.

Perhaps it goes without saying… as you read this note, I appeal to you to ask questions and make comments and suggestions. Thank you. –Bruce Camber

Planck Time is the smallest possible unit of measurement of time. The ratios of all 201+ multiples of the Planck Time to its respective multiple of Planck Length is consistent across the chart.  The original calculations were done by Max Planck in and around 1899. This chart of 201+ notations was done in December 2014. Any numbers smaller than the Planck Time are just numbers that cannot be meaningfully applied to anything.

Planck got his Nobel Prize in 1918 for his discovery of energy quanta. He was also a mentor and friend of Einstein (who received his Nobel Prize in 1921).

The Planck Length and Planck Time are actual values that can be multiplied by 2.
Of course, if one were to multiply each by 2 over and over again, you can assume that you would reach their outer limits. That process looks a bit tedious. After all, the Age of the Universe is somewhere over 13.8+ billion years and the Observable Universe is millions of light years from common sense. Yet, rather surprisingly, to complete that effort doesn’t require thousands of doublings. It is done in somewhere just over 201+ doublings.

That is so surprising, the doublings for both are charted below.

These doublings do kind-of, sort-of end up in sync. Where there is a problem, we assume it is within our simple math. Considering the duration and the length, and the nature of very large measurements, for all intents and purposes, they are synced mathematically. We’ve got a bit of work to do to sync them up intellectually!

Though these charts will be tweaked substantially, the best place to start is at the notations (or doublings) that define a day, a week and a year (in Planck Time units) to see how each corresponds with the distance light travels in Planck Length units, i.e. a light year, “light week,” and “light day.” These are our first baby steps of analysis. How many hundreds of steps are there to go to discern all the faces of its meaning? Who knows? From here, we will continue to look to see what meaning and relation evolves at a particular notation where one column appears to impart value to the other. Just on the surface, this chart seems to suggest that there are other possible views of the nature of space and time where order (sequence), continuity, symmetries, and relations seem to play a more fundamental role.

Science and our common sense worldview assume the primordial nature of space and time. As a result of our work with the Planck Units, we hold that conclusion up for further inspection. How do things appear as one begins to approach a synchronized Planck Length and Planck Time?

Planck Units: As we add more Planck Units to this chart, what else might we see? What might we learn? So, we will add mass, electric charge, and temperature to these listings. And then, we’ll add the derived Planck Units (12) and then ask, “Is there anything more we can do to establish a range from the smallest to the largest? What might a comparative analysis at each doubling reveal to us?” We don’t know, however, we are on a path to explore! We’ll report in right here.

At this point, we are attempting to learn enough to make a few somewhat educated guesses about the nature of things within these scales of the universe.

So, as a result of where we are today, I think it is okay to ask the question, “What would the universe look like if space and time were derivative of order-continuity and relation-symmetry, and of ratios where the subject-object are constantly in tension?”

This stream of consciousness continues at the very bottom of this chart.

Planck Time Doublings:
Primarily in Seconds
Planck Length Doublings:
Primarily in Meters
204

The Age of the Universe:
13.78 to 13.8 billion years

Observable Universe: 8.8×10(26) m  Planck Multiple: 8.31×1026 m
203 It appears that we currently live in the earliest part of 201 doubling: 4.155×1026 m Future Universe
202 6.9309178×1017 seconds (21.9777+ billion years) 2.077×1026 m Future Universe
201 346,545,888,147,200,000 seconds (10.9888+ billion years) 1.03885326×1026 m  Observable Universe
In this model: Time is discrete so to know how many years are to be aggregated (to see how close we are to the Age of the Universe), each notation must be added together. By the 200th notation, we would be one Planck Time unit shy of 10.9888 billion years. A possible conclusion could therefore be that we are within the 201st notations.
200 173,272,944,073,600,000 seconds (5.49444+ billion years) (1017) 5.19426632×1025 m
199 86,636,472,036,800,000 seconds (2.747+ billion years) 2.59713316×1025 m
198 43,318,236,018,400,000 seconds (1.3736+ billion years) 1.29856658×1025 m
197 21,659,118,009,200,000 seconds (686.806+ million years) 6.49283305×1024 m
196 10,829,559,004,600,000 seconds (342.4+ million years) (1016) 3.24641644×1024 m
195 5,414,779,502,320,000 seconds (171.2+ million years) 1.62320822×1024 m
194 2,707,389,751,160,000 seconds (85.6+ million years) 8.11604112×1023 m
193 1,353,694,875,580,000 seconds (42.8+ million years) (1015) 4.05802056×1023 m
192 676,847,437,792,000 seconds (21.4+ million years) 2.02901033×1023 m
191 338,423,718,896,000 seconds (10.724+ million years) 1.01450514×1023 m
190 169,211,859,448,000 seconds (5.3+ million years) (1014) 5.07252568×1022 m
189 84,605,929,724,000 seconds (2.6+ million years) 2.5362629×1022 m
188 42,302,964,862,000 seconds (1.3+ million years) 1.26813145×1022 m
187 21,151,482,431,000 seconds (640+ thousand years) 6.34065727×1021 m
186 10,575,741,215,500 seconds (320+ thousand years) (1013) 3.17032864×1021 m or 3 Zettameters or 310,000 ly
185 5,287,870,607,760 seconds (160+ thousand years) 1.58516432×1021 m or about 150,000 ly (1.5z)
184 2,643,935,303,880 seconds (83.7+ thousand years) 7.92582136×1020 m
183 1,321,967,651,940 seconds (41.8+ thousand years) (1012) 3.96291068×1020 m
182 660,983,825,972 seconds (20.9+ thousand years) 1.981455338×1020 m
181 330,491,912,986 seconds (or about 10,472.9 years) 9.90727664×1019 meters
180 165,245,956,493 seconds (1011) 4.95363832×1019 m
179 82,622,978,246.4 seconds 2.47681916×1019 m
178 41,311,489,123.2 seconds 1.23840958×1019 m
177 20,655,744,561.6 seconds 6.19204792×1018 m
176 10,327,872,280.8 seconds (1010) 3.09602396×1018 m
175 5,163,936,140.4 seconds 1.54801198×1018 m
174 2,581,968,070.2 seconds 7.74005992×1017 m
173 1,290,984,035.1 seconds (109) 3.87002996×1017 m
172 645,492,017.552 seconds 1.93501504×1017 m
171 322,746,008.776 seconds 9.67507488×1016 m
170 161,373,004.388 seconds (108) 4.83753744×1016 m
169 80,686,502.194 seconds 2.41876872×1016 m
168 40,343,251.097 sec (466 days)(Note: 31,536,000 s/year) 1.20938436×1016 m
Comments: A light year is about 9.4605284×1015 meters (Google) or 9,460,730,472,580,800 metres “exactly” (Wikipedia). Use the Gregorian calendar (circa 1582) where a year is 365.2425 and the speed of light is given as 299,792,458 metres/second, the calculation is 365.2425 times 86400 seconds/day (or 31556952 seconds/year) times 299,792,458 meters/second or 9.4605362+×1015 meters. Discrepancies would become quite large at the size of the Observable Universe and the Age of the Known Universe.Using Planck Units:
One Light Year 9.45994265715×1015m
167 20,171,625.5485 seconds (233.468 days) 6.0469218×1015 m [one light year (ly) is 9.4×1015 m]
166 10,085,812.7742 seconds (116.73 days) (107) 3.0234609×1015 m
165 5,042,906.38712 seconds (58.36+) 1.5117305×1015 m
164 2,521,453.19356 s (29.1835 days) 7.55865224×1014 m
163 1,260,726.59678 s (14.59+ days) (106) 3.77932612×1014 m
162 630,363.29839 s (7.29+ days) 1.88966306×1014 m (about 7-day light travel)
161 315,181.649195 seconds (3.64794 days) 9.44831528×1013 m
160 157,590.824 s (1.82 days) (105) 4.72415764×1013 m
159 78,795.4122988 s (.911984 days) 2.36207882×1013 m (or close to 24-hour light travel)
158 39,397.7061494 seconds 1.18103945×1013 m
157 19,698.8530747 seconds (104) 5.90519726×1012 m
156 9849.42653735 seconds 2.95259863×1012 m ()
155 4924.71326867 seconds(3600 s in hour) 1.47629931×1012 m
154 2462.35663434 seconds 738,149,657 kilometers 1011
153 1231.17831717 seconds (103) 369,074,829 kilometers 1011
152 615.589158584 seconds (10.259+ minutes) 184,537,414 kilometers 1011
151 307.794579292 seconds 92,268,707.1 kilometers (range of earth-to-sun)1010m
150 153.897289646 seconds (102) 46,134,353.6 kilometers 1010
149 76.948644823 s (16+ sec over 1 min) 23,067,176.8 kilometers  1010
Comments: A light minute is, of course, sixty times 299,792.458 km/second. Again, using simple mathematics, the distance light travels in one minute is 17,987,547.48 which is about 1000 kilometers off of 17,986,420.0329 km/second using the simple mathematics of this chart. This difference will be further analyzed.
148 38.4743224115 s (21.53 sec to 1 min) 11,533,588.4 kilometers  1010
147 19.2371612058 seconds 101 5,766,794.2 kilometers 109
146 9.61858060288 seconds 2,883,397.1 kilometers 109
145 4.80929030144 seconds 1,441,698.55 kilometers 109 m
144 2.40464515072 seconds 720,849.264 kilometers 108
143 1.20232257536 s (1s ≠ perfect tp multiple)
One Second:
360,424.632 kilometers 108 meters
Speed of light equals 299,792,458 m/s
Comments: Science knows experimentally that light travels 299,792.458 km/second (a light second). A Planck Time multiple, either 1.202 seconds or .6011 seconds, could be used as a standard unit of time that is based on a theoretical constant. We will explore further the calculations for a day, week, month and year based on such a system. We’ll also explore it in light of recent work to define the theoretical chronon.
A Light Second 299,792.458 km
142 6.0116128768×10−1 seconds 180,212.316 kilometers (111,979+ miles) 108 m
141 3.0058064384×10−1 seconds 90,106.158 kilometers 107 m
140 1.5029032192×10−1 seconds 45,053.079 kilometers 107
139 7.514516096×10−2 seconds 22,526.5398 kilometers 107
138 3.757258048 × 10−2 seconds 11,263.2699 kilometers or about 7000 miles
137 1.878629024 × 10−2 seconds 5631.63496 kilometers 106
136 9.39314512 × 10−3 seconds 2815.81748 kilometers 106

The transition from the Human-Scale to the Large-Scale Universe

135 4.69657256 × 10−3 seconds 1407.90874 kilometers (about 874 miles) 106 m
134 2.34828628 × 10−3 seconds 703.954368 kilometers 105
133 1.174143145978 × 10−3 seconds 351.977184 kilometers (218.7 miles) 105
132 5.8707157335 × 10−4 seconds 175.988592 kilometers (109.35 miles) 105
131 2.93535786675 × 10−4 seconds 87.994296 kilometers 104
130 1.46767893338 × 10−4 seconds 43.997148 kilometers 104
129 7.33839466688 × 10−5 seconds 21.998574 kilometers104
128 3.66919733344 × 10−5 seconds 10.999287 kilometers or within 6.83464 miles 104
127 1.83459866672× 10−5 seconds 5.49964348 kilometers 103
126 9.1729933336 × 10−6 seconds 2.74982174 kilometers 103
125 4.5864966668 × 10−6 seconds 1.37491087 kilometers 103
124 2.2932483334 × 10−6 seconds 687.455439 meters 102
123 1.1466241667 × 10−6 seconds 343.72772 meters or about 1128 feet 102
122 5.73312083348 × 10−7 seconds 171.86386 meters or about 563 feet 102
121 2.86656041674 × 10−7 seconds 85.9319296 meters 101
120 1.43328020837 × 10−7 s 42.9659648 meters 101
119 7.16640104186 × 10−8 sec 21.4829824 meters 101
118 3.58320052093 × 10−8 sec 10.7414912 meters or 35.24 feet or 1.074×101 m 101
117 1.79160026046 × 10−8 seconds 5.3707456 meters 100
116 8.95800130232 × 10−9 seconds 2.6853728 meters or 105.723 inches 100
115 4.47900065116 × 10−9 seconds 1.3426864 meters or 52.86 inches 100
114 2.23950032558 × 10−9 seconds 67.1343176 cm (19.68+ inches or 6.71×10-1
113 1.11975016279 × 10−9 seconds 33.5671588 centimeters or 3.356×10-1 m)
112 5.59875081396 × 10−10 seconds 16.7835794 centimeters or 1.6783×10-1
111 2.79937540698 × 10−10 seconds 8.39178968 cm (3.3+ inches or 8.39×10-2 m)
110 1.39968770349 × 10−10 seconds 4.19589484 centimeters 4.19589484×10-2 m
109 6.99843851744 × 10−11 seconds 2.09794742 centimeters or 2.0979×10-2 m
108 3.49921925872 × 10−11 seconds 1.04897 centimeters or 1.04897375×10-2 m
107 1.74960962936 × 10−11 seconds 5.24486856 mm (about 1/4 inch) or 5.24×10-3 m
106 8.7480481468 × 10−12 seconds 2.62243428 millimeters or 2.62243428×10-3 m
105 4.3740240734 × 10−12 seconds 1.31121714 millimeters 1.31121714×10-3 m
104 2.1870120367 ×10−12 seconds .655608568 millimeters or 6.55608568×10-4 m
103 1.09350601835 ×10−12 seconds .327804284 millimeter or 3.27804284 x10-4 m
102 5.46753009176 ×10−13 seconds .163902142 millimeters or 1.63902142×10-4 m
101 2.73376504588 × 10−13 seconds 81.9510712 microns or 81.9510712 x10-5 m
100 1.36688252294 × 10−13 seconds 40.9755356 microns or 4.09755356 x10-5 m
99 6.83441261472 × 10−14 seconds 20.4877678 microns or 2.04877678×10-5 m
98 3.41720630736 × 10−14 seconds 10.2438839 microns or 1.02438839×10-5 m
97 1.70860315368 × 10−14 seconds 5.12194196 microns (.0002+ inches or 5.12×10-6 m)
96 8.5430157684 × 10−15 seconds 2.56097098 microns or 2.56097098×10-6 m
95 4.2715078842 × 10−15 seconds 1.28048549 microns or 1.2804854×10-6 m
94 2.1357539421 × 10−15 seconds 640.242744 nanometers 6.40242744×10-7m
93 1.06787697105 × 10−15 seconds 320.121372 nanometers 3.20121372×10-7 m
92 5.33938485524 × 10−16 seconds 160.060686 nanometers or 1.6×10-7 m
91 2.66969242762 × 10−16 seconds 80.0303432 nanometers or 8.0×10-8 m
90 1.33484621381 × 10−16 seconds 40.0151716 nanometers or 4.0×10-8 m
89 6.67423106904 × 10−17 seconds 20.0075858 nanometers or 2.0×10-8 m
88 3.33711553452 × 10−17 seconds 1.00037929×10-8 meters or 10 nanometers
87 1.66855776 × 10−17 seconds (smallest measurement – 2010) 5.00189644×10-9 meters
86 8.34278883632 × 10−18 seconds 2.50094822 nanometers or 2.50094822×10-9 m
85 4.17139441816 × 10−18 seconds 1.25474112 nanometers or 1.25×10-9 m
84 2.08569720908 × 10−18 seconds .625237056 nanometers or 6.25237056×10-10 m
83 1.04284860454 × 10−18 seconds .312618528 nanometers or 3.12×10-10 m
82 5.21424302272 × 10−19 seconds .156309264 nanometers or 1.563×10-10 m
81 2.60712151136 × 10−19 seconds 7.81546348×10-11 m
80 1.30356075568 × 10−19 seconds 3.90773174×10-11 m
79 6.5178037784 × 10−20 seconds 1.95386587×10-11 m
78 3.2589018892 × 10−20 seconds 9.76932936×10-12 m
77 1.6294509446 × 10−20 seconds 4.88466468×10-12 m
76 8.147254723 × 10−21 seconds 2.44233234×10-12 m
75 4.0736273615 × 10−21 seconds 1.22116617×10-12 m
74 2.03681368075 × 10−21 seconds 6.10583084×10-13 m
73 1.01840684038 × 10−21 seconds 3.05291542×10-13 m
72 5.09203420188 × 10−22 seconds 1.52645771×10-13 m
71 2.54601710094 × 10−22 seconds 7.63228856×10-14 m
70 1.27300855047 × 10−22 seconds 3.81614428×10-14 m
69 6.36504275236 × 10−23 seconds 1.90807214×10-14 m
68 3.18252137618 × 10−23 seconds 9.54036072×10-15 m

Transition from the Small-Scale Universe to the Human-Scale Universe

67 1.59126068809 × 10−23 seconds 4.77018036×10-15 m
66 7.95630344044 × 10−24 seconds 2.38509018×10-15 m
65 3.97815172022 × 10−24 seconds 1.19254509×10-15 m
64 1.98907586011 × 10−24 seconds 5.96272544×10-16 m
63 9.94537930056 × 10−25 seconds 2.98136272×10-16 m
62 4.97268965028 × 10−25 seconds 1.49068136×10-16 m
61 2.48634482514 × 10−25 seconds 7.45340678×10-17 m
60 1.24317241257 × 10−25 seconds 3.72670339×10-17 m
59 6.21586206284 × 10−26 seconds 1.86335169×10-17 m
58 3.10793103142 × 10−26 seconds 9.31675848×10-18 m
57 1.55396551571 × 10−26 seconds 4.65837924×10-18 m
56 7.76982757856 × 10−27 seconds 2.32918962×10-18 m
55 3.88491378928 × 10−27 seconds 1.16459481×10-18 m
54 1.94245689464 × 10−27 seconds 5.82297404×10-19 m
53 9.7122844732 × 10−28 seconds 2.91148702×10-19 m
52 4.8561422366 × 10−28 seconds 1.45574351×10-19 m
51 2.4280711183 × 10−28 seconds 7.27871756×10-20 m
50 1.21403555915 × 10−28 seconds 3.63935878×10-20 m
49 6.07017779576 × 10−29 seconds 1.81967939×10-20 m
48 3.03508889788 × 10−29 seconds 9.09839696×10-21 m
47 1.51754444894 × 10−29 seconds 4.54919848×10-21 m
46 7.58772224468 × 10−30 seconds 2.27459924×10-21 m
45 3.79386112234 × 10−30 seconds 1.13729962×10-21 m
44 1.89693056117 × 10−30 seconds 5.68649812×10-22 m
43 9.48465280584 × 10−31 seconds 2.84324906×10-22 m
42 4.74232640292 × 10−31 seconds 1.42162453×10-22 m
41 2.37116320146 × 10−31 seconds 7.10812264×10-23 m
40 1.18558160073 × 10−31 seconds 3.55406132×10-23 m
39 5.92790800364 × 10−32 seconds 8.88515328×10-24m
38 2.96395400182 × 10−32 seconds 4.44257664×10-24 m
37 1.48197700091 × 10−32 seconds 2.22128832×10-24m
36 7.40988500456 × 10−33 seconds 1.11064416×10-24m
35 3.70494250228 × 10−33 seconds 5.5532208×10-25m
34 1.85247125114 × 10−33 seconds 2.7766104×10-25m
33 9.26235625568 × 10−34 seconds 1.3883052×10-25m
32 4.63117812784× 10−34 seconds 6.94152599×10-26m
31 1.15779453196× 10−34 seconds 3.47076299×10-26m
30 5.78897265978 × 10−35 seconds 1.735381494×10-26 m
29 2.89448632989 × 10−35 seconds 8.67690749×10-27 m
28 1.44724316494 × 10−35 seconds 4.3384537×10-27 m
27 7.23621582472 × 10-36 seconds 2.16922687×10-27 m
26 3.61810791236 × 10−36 seconds 1.0846134×10-27 m
25 1.80905395618 × 10−36 seconds 5.42306718×10-28 m
24 9.045269781089 × 10−37 seconds 2.711533591×10-28 m
23 4.522263489044 × 10−37 seconds 1.35576679×10-28 m
22 2.26131744522 × 10−37 seconds 6.77883397×10-29 m
21 1.13065872261 × 10−37 seconds 3.38941698×10-29 meters
20 5.65329361306 × 10−38 seconds 1.69470849×10-29 meters
19 2.82646806528 ×10−38 seconds 8.47354247×10-30 meters
18 1.41323403264 ×10−38 seconds 4.2367712×10-30 m
17 7.0661701632 × 10−39 seconds 2.11838561×10-30 m
16 3.530850816 × 10−39 seconds 1.0591928×10-30 m
15 1.7665425408 × 10−39 seconds 5.29596404×10-31 m
14 8.832712704 × 10−40seconds 2.64798202×10-31 m
13 4.416356352 × 10−40 seconds 1.32399101×10-31 m
12 2.208178176 × 10−40 seconds 6.619955ƒx10-32 m
11 1.104089088 × 10−40 seconds 3.30997752×10-32 m
10 5.52044544 × 10−41 seconds 1.65498876×10-32 m
9 2.76022272 × 10−41 seconds 8.27494384×10-33 m
8 1.38011136 × 10−41 seconds 4.1374719232×10-33 m
7 6.9005568 × 10−42 seconds 2.0687359616×10-33 m
6 3.4502784 × 10−42 seconds 1.03436798×10-33 m
5 1.7251392 × 10−42 seconds 5.1718399×10-34 m
4 8.625696 × 10−43 seconds 2.58591995×10-34 m
3 4.312848 × 10−43 seconds 1.29295997×10-34 m
2 2.156424 × 10−43 s The second doubling 6.46479988×10-35 meters
1 1.078212 × 10−43 s The first doubling 3.23239994×10-35 m The first doubling, step, or layer.
5.39106(32)×10−44 seconds 1.616199(97)x10-35 meters

The Planck Time

The Planck Length

Endnotes:1. We are in the process of refining this chart and will be throughout 2015 and 2016.

2. Our very first calculation with the Planck Length column (December 2011), resulted in 209 doublings! We found several errors. Then , with help of a NASA astrophysicist, Joe Kolecki (now retired), we updated our postings with his calculation of 202.34. Then, a French Observatory astrophysicist, Jean-Pierre Luminet, calculated 205.1 doublings. We are very open to all ideas and efforts! We are studying the foundations of foundations. One might call it a hypostatic science based on the simplest mathematics, simple geometries and observations about the way the universe coheres.

One might say, “The Finite is finite, the Infinite is the Infinite, and the constants and universals describe the boundary conditions and transformations between each. One manifests a panoply of perfections; the other has only momentary instants of perfection.”

Open Questions:  Is it possible that Planck Length starts first and Planck Time begins on Planck Length’s 2nd or 3rd notation? Could there be a two for one at the beginning and at the transitions?   By using experimental speed of light per second, can we force the Planck units from that point?    If the ratio of Length/time is consistent across the grid (and it is), how do we fine tune this chart?

What is a second?

What are Planck Units?

What is time?

What is a meter?

What is length?

What is space?

Notes
1.  What happens just before the Planck time at 10-44 seconds? Theorists say that all of the four fundamental forces are presumed to have been unified into one force. All matter, energy, space and time “explode” from the original singularity.

2. On May 10, 2010, the very smallest unit of measured time was experimentally demonstrated; the result was 1.2 × 10−17 seconds. That is a long way from 10−44 seconds! For more background, see: http://phys.org/news192909576.html

3. Our online “Google” calculator often rounds up the last digit. It is usually beyond the eleventh position to the right of the decimal point.

4. For more about this place and time, go to Hyperphysics (Georgia State): http://hyperphysics.phy-astr.gsu.edu/hbase/astro/planck.html

5. A copy of this chart has also been published in the following locations:

a. http://walktheplanck.wordpress.com/2014/12/09/base/

b. http://utable.wordpress.com/2014/12/12/planck/

c. http://SmallBusinessSchool.org/page3053.html

d. ResearchGate Documents: 3052, 3054, 3056

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