Last Update: Monday June 21, 2021

# Notations from Planck's length to Notation #1, to Notation #206, at the edge of the observable universe (this is page 1 of 4)

 This discussion extends the general overview of the Big Board-little universe  and a rough-draft for an article about it all. Of course, even this page is a work-in-progress. What is simple? A point? What do we know about a point?  There are so many concepts to learn just in 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 begins at the Planck length and simply multiplies it by two over and over again.. Powers-of-two and Exponentiation based on the Planck length.  Herein it is referred to as Base-2 Exponential Notation, Powers of Two based on the Planck length"   (abbreviated,  "B2EN").  Can the universe, from the smallest to the largest, be seen in a more meaningful way than using base-ten scientific notation (B10SN)?  B2EN renders greater granularity,  a necessary relationality, and nesting geometries.   The project originated with a series of five high school geometry classes in December 2011. The smallest possible measurement within space and time was defined mathematically back in 1889 by a fellow named Max Planck. Generally accepted within the scientific community, the Planck length is so small, it is written using scientific notation. The number is 1.616199(97)x10-35 meters.  As a starting point we looked for all 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, those references will grow substantially.  As we understand it today, we believe it must  be one of the most 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 number can be used to define a point. 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. Mathematically, it will require just 202.34 notations. In five columns,  the first column is the base-ten notations. The second column is a Planck number based on the number of times the Planck length has been multiplied. The third column is the number of points, 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 about a notation. B10SN 1 B2EN 0 Number of points 2^0=1 Length (width-height)meters 1.616199(97)×10-35m Additional Information, Discussion, Examples: At this first notation, there is just one point, a singularity. More... 1 1 2^1=2 3.23239994×10-35m At the second notation, there are two points, the shortest possible line, possibly the beginnings of a string which opens discussions about some of the most basic questions in science. 1 2 2^2=4 6.46479988×10-35m At the third notation there are four points. There are several logical possibilities: (1) four points form a line, (2) four points form a jagged line of which four [[skew polygon|skewed triangles]] could be formed, (3) three points form a triangle that define a plane with the fourth point forming a [[tetrahedron]] (imperfect or perfect) that defines the first three dimensions of space 2 3 2^3=8 1.292959976×10-34m At the fourth notation there are eight points. The logical possibilities are now expanded to include placing the points 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 points are are equally distributed on the edges of tetrahedron, an octahedron and four tetrahedrons begin to emerge. (picture to be added) If added within, [[Close-packing of equal spheres|tetrahedrally close-packed structures]] emerge. If added externally, with just three additional points, a tetrahedral pentagon is created of five tetrahedrons. (picture to be added) With all eight additional points added externally,a [[cube]] or [[hexahedron]] could be created. 2 4 2^4=16 2.585919952×10-34m At the fifth notation there are sixteen points. If any one of the points were to become a center point, and 10 points are extended from it, a tetrahedral icosahedron emerges. (picture to be added) With twenty points a simple dodecahedron is possible. And with the icosahedron, all [[platonic solids|five basic 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 points. (picture to be added) These points 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 point of the octahedron would begin to be defined and by the 18th point 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) 2 5 2^5=32 5.171839904×10-34m At the sixth notation, there are 32 points. Here there is a possibility for a cluster of eight tetrahedral pentagons to emerge and complete with 34 points. 3 6 2^6=64 1.0343679808×10-33m At the seventh notation, there are 64 points. With just 43 of those points a hexacontagon could be created. It has 12 polytetrahedral clusters with an icosahedron and many tetrahedrons in the middle. 3 7 2^7=128 2.0687359616×10-33m By the eighth notation, the progression becomes self-evident. The results are not. Simple exponential notation based on the power of two is well documented. Of course, by using the Power of two, exponentiation and starting at Planck's constant, necessary relations can be constructed. 3 8 2^8=256 4.1374719232×10-33m Geometric complexification to be discussed. 3 9 2^9=512 8.2749438464×10-33m _ 4 10 1024 1.65498876928×10-32m _ 4 11 2048 3.30997752836×10-32m _ 4 12 4096 6.61995505672×10-32m _ 5 13 8192 1.323991011344×10-31m _ 5 14 16,384 2.647982022688×10-31m _ 5 15 32,768 5.295964045376×10-31m _ 6 16 65,536 1.0591928090752×10-30m _ 6 17 131,072 2.1183856181504×10-30m _ 6 18 262,144 4.2367712363008×10-30m _ 6 19 524,288 8.4735424726016×10-30m _ 7 20 1,048,576 1.69470849452032×10-29m _ 7 21 2,097,152 3.38941698904064×10-29m more information 7 22 4,194,304 6.77883397808128×10-29m _ 8 23 8,388,608 1.355766795616256×10-28m _ 8 24 16,777,216 2.711533591232512×10-28m _ 8 25 33,554,432 5.423067182465024×10-28m _ 9 26 67,108,864 1.0846134364930048×10-27m _ 9 27 134,217,728 2.1692268729860096×10-27m 9 28 268,435,456 4.3384537459720192×10-27m _ 9 29 536,870,912 8.6769074919440384×10-27m _ 10 30 1,073,741,824 1.73538149438880768×10-26m _ 10 31 2,147,483,648 3.47076299879961536×10-26m _ 10 32 4,294,967,296 6.94152599×10-26m _ 11 33 8,589,934,592 1.3883052×10-25m _ 11 34 1.7179869×1011 2.7766104×10-25m Actual number: 17,179,869,184 points 11 35 3.4359738×1011 5.5532208×10-25m 34,359,738,368 12 36 6.8719476×1011 1.11064416×10-24m 68,719,476,736 12 37 1.3743895×1012 2.22128832×10-24m 137,438,953,472 12 38 2.7487790×1012 4.44257664×10-24m 274,877,906,944 12 39 5.4975581×1011 8.88515328×10-24m 549,755,813,888 13 40 1.0995116×1012 1.77703066×10-23m 1,099,511,627,776 13 41 2.1990232×1012 3.55406132×10-23m 2,199,023,255,552 13 42 4.3980465×1012 7.10812264×10-23m 4,398,046,511,104 14 43 8.7960930×1012 1.4.2162453×10-22m 8,796,093,022,208 14 44 1.7592186×1013 2.84324906×10-22m 17,592,186,044,416 14 45 3.5184372×1013 5.68649812×10-22m 35,184,372,088,832 15 46 7.0368744×1013 1.13729962×10-21m 70,368,744,177,664 15 47 1.4073748×1014 2.27459924×10-21m 140,737,488,355,328 15 48 2.8147497×1014 4.54919848×10-21m 281,474,976,710,656 15 49 5.6294995×1014 9.09839696×10-21m 562,949,953,421,312 16 50 1.12589988×1015 1.81967939×10-20m 1,125,899,906,842,624 16 51 2.25179981×1015 3.63935878×10-20m 2,251,799,813,685,248 16 52 4.50359962×1015 7.27871756×10-20m 4,503,599,627,370,496 17 53 9.00719925×1015 1.45574351×10-19m 9,007,199,254,740,992 17 54 1.80143985×1016 2.91148702×10-19m 18,014,398,509,481,984 17 55 3.60287970×1016 5.82297404×10-19m 36,028,797,018,963,968 18 56 7.205759840×1016 1.16459481×10-18m 72,057,594,037,927,936 18 57 1.44115188×1017 2.32918962×10-18m 144,115,188,075,855,872