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Cs184 1x Homework 13

About CS184.1x

CS184.1x teaches the Foundations of Computer Graphics. Students will understand the concepts of 3D graphics, and develop programs that create images of a 3D scene with lighting, using both real-time OpenGL and GLSL programming, as well as offline raytracing. The focus is on foundations and writing your own programs, rather than learning use of a specific software like Maya or Photoshop.

Course Overview

You can check out the syllabus page to see the topics we will be covering, along with lecture slides, videos and homework assignments. Briefly, the course is organized in 4 units, each with its own homework assignment.

  1. Overview and Basic Math (Homework 0: 10% of grade)
  2. Transformations (Homework 1: 20% of grade)
  3. OpenGL and Lighting (Homework 2: 35% of grade)
  4. Raytracing (Homework 3: 35% of grade).

Units/homeworks 0 and 1 are allotted 1 week each, while the more challenging homeworks 2 and 3 are allotted 2 weeks each. The full course is 6 weeks, and should provide a solid foundation for interactive and offline 3D computer graphics.

This term, students who earn a total score of 50% or greater will have passed the course and may obtain a free honor code certificate from BerkeleyX. As with all EdX classes, no numerical score or grade will be shown on the certificate.

Video Lectures

The video lectures are available from the courseware and hosted on YouTube. Since YouTube is not available in some countries, we also provide links for each lecture video, to download the mp4 video, slides and transcript, as well as half-resolution video for lower bandwidth. Some students who want to download the videos for offline use have also reported that lower filesize videos can be obtained using keepvid. On many lectures, I will derive equations or code, writing on the slides (I use a WaCom Cintiq 12WX tablet attached to a MacBook Pro). I believe this keeps the lecture engaging, as opposed to purely watching PowerPoint. However, I admit that my handwriting is not the best. For any relevant mathematical equation, the full typeset equation will always be shown, usually on the immediately next slide. The lecture slides PDFs you can download also contain all of the equations. The EdX video viewer should support interactive transcripts, and resolutions from 240p to HD 720p. In case you want a direct link to the YouTube videos, you can access the local UC Berkeley website for Online Lectures, as well as directly browse the CS 184 YouTube channel.  Finally, please note that there are multiple video segments for each lecture, and you can access them off the tabs on the top.  For convenience, the slides include the full lecture (all segments).

Prerequisites

  • C++ (knowledge of C or Java should be adequate to learn C++)
  • A computer (laptop or desktop) capable of running Homework 0
  • Basic math: vectors and matrices (a review is given in Lecture 2)
The course makes extensive use of OpenGL (and C++) in homeworks 0,1 and 2, mainly as a tool/language to understand and implement graphics concepts. While no prior knowledge of OpenGL is required, you will be expected to learn the basics and program in it largely from online documentation; learning/using a new language from searching online documentation is not unusual for experienced programmers.

Late Policy

Assignments are due by the due date listed on the syllabus at 11:59pm Pacific Standard Time (PST is GMT -8 hours). We provide a grace period (with no penalty) of 3 days to account for unexpected issues. Beyond that point, no credit will be given, and the graders will not work after that point.

We recommend you to start early on the homework assignments. Experiences with the local Berkeley class tells us that the homework 1 should take around 12 hours, and homeworks 2 and 3 should take about 24 hours. Your mileage may vary (one student in the Fall EdX class commented that time for homework completion increases geometrically as about 1, 3, 9, 27 hours for homeworks 0,1,2,3), but please plan ahead.

Textbooks and Computer Requirements

There are no textbooks required; the course is free. We post links to some free OpenGL and GLSL tutorials; you can search online for many others. You may use any computer system that supports programmable shaders on a graphics card (GPU). Most computers (including laptops) built in the last 5-10 years satisfy this requirement.  More precisely, your graphics card must support at least GLSL version 120 (OpenGL 2.1).  This is true for almost any NVIDIA or ATI GPU, but some old integrated graphics cards (Intel GMA 950 or similar) may not run the homeworks appropriately.  Given the variety of systems, this should only be taken as a guideline; you can test your machine by doing homework 0.

Grading

Grading will be based on the four programming projects.

Honor Code

You may discuss your homework and class material with classmates; in fact, we encourage you to do so (please avoid spoilers and explicit code on the forums, though!). However, all submitted work must be your own, and you must write all programs yourself (no copying code from classmates, previous instances of the class or online resources). You must only submit images for grading that were actually produced by your program. You must abide by the edX Honor Code. Please do not post your source code or programs on external websites or social media. In particular, do not post source code to a public repository on github or a similar site; use private/not searchable repositories, or simply do not use github.  

Communication

Announcements will be posted in the updates & news section. Our primary communication channel will be our interactive forum, where students and staff will all contribute to an ongoing discussion of the course material.

Course staff

Ravi Ramamoorthi

Ravi Ramamoorthi was a Professor at UC Berkeley. He has moved to UCSD since Jul 1, 2014, but continues to maintain an affiliation at Berkeley; this course was developed and is still being offered as a BerkeleyX class. He has taught computer graphics more than 10 times at Stanford, Columbia and UC Berkeley, and has been honored with a number of awards for his research, including the ACM SIGGRAPH Significant New Researcher Award and by the White House with the PECASE (Presidential Early Career Award for Scientists and Engineers).

Course Technical Staff

Berkeley students Nicholas Estorga and Brandon Wang have done a tremendous job developing the auto-graders and other materials used in the course, but have now graduated. Current Berkeley PhD student Ling-Qi Yan will be maintaining the graders for this iteration.  A number of community TAs from previous local and online iterations of the course will also be answering questions on the forums.  

Received Date: Mar 04, 2014 / Accepted Date: May 20, 2014 / Published Date: Jun 05, 2014

When studying the genetic code, the first question to ask could be: “What is a ‘code’?” The most obvious answer is that a code is a set of correspondences between two different languages. In the same way, the bioinformatics context of this article is an interface between the geometry within bioinformatics and its cryptography [1].

For example, let’s remember the intuitive letter-to-number code of our childhood (evocative of the Roman numerals that lacked of zero), also called the “letter number” cipher or code, in which the letters of an alphabet are replaced by numbers in their respective order: transforming abc… into 1-2-3…1

A more counter-intuitive variant of such letter-to-number code is the one that includes the zero (0) at the start (as did the Arabic numerical system, now in worldwide use, even in programming, a system that apparently was developed independently by the Mayans); starting with abc…, which in this new code that includes the zero, is transformed into 0-1-2…, ending with …xyz being translated into …23-24-25 [1]; and, even the fingers of both of our hands can be used in mnemonics in order to remember this code, and by analogy, to remember the genetic code; furthermore, our four extremities can be used to remember its four foundational nucleotides (A, T, C, G)2 for a better understanding of the bio-flexibility of the genetic code.

Thus far, these are the basic aspects on the study of the genetic code, aspects relatively easy to remember by a student exposed for the first time to it; but we are going to find here additional tools that can be used, not only in education, but also in bioinformatics, for the more deeply appreciation of the source of information contained within the genetic code. For the basic computational concept of file compression, we read: “File compression reduces the size of a file by cleverly taking out parts of the contents of the file that aren’t needed... Zipping is very common, particularly because it reduces the amount of data that needs to be transported from here to there” [2].

We need to start with the unpublished and handwritten representation of the set of 64 codons done by Nirenberg in 1965 [3] (shown in Table 1; published in full as a primer in this article). After the representation of Nirenberg (Table 1), the table of Crick arrived in 1967 [4], then the circular representation by Bresch and Hausmann in 1972 [5], and then Fujimoto’s tetrahedron in 1987 [6], being the last one similar to a backbone developed by Kepler many years before [7]; then, a set of more recent rules of variation [8,9] was discovered while carefully analyzing [5], plus newer representations, including the use of the I Ching [10], the tetrahedron [11], and the rotating square [12], among others.

  1234
  UCAG
1UU5’ – UUU = F AAA – 5’5’ – UUC = F AAG – 5’5’ – UUA = L AAU – 5’5’ – UUG = L AAC – 5’
2UC5’ – UCU = S AGA – 5’5’ – UCC = S AGG – 5’5’ – UCA = S AGU – 5’5’ – UCG = S AGC – 5’
3UA5’ – UAU = Y AUA – 5’5’ – UAC = Y AUG – 5’5’ – UAA = * AUU – 5’5’ – UAG = * AUC – 5’
4UG5’ – UGU = C ACA – 5’5’ – UGC = C ACG – 5’5’ – UGA = * ACU – 5’5’ – UGG = W ACC – 5’
5CU5’ – CUU = L GAA – 5’5’ – CUC = L GAG – 5’5’ – CUA = L GAU – 5’5’ – CUG = L GAC – 5’
6CC5’ – CCU = P GGA – 5’5’ – CCC = P GGG – 5’5’ – CCA = P GGU – 5’5’ – CCG = P GGC – 5’
7CA5’ – CAU = H GUA – 5’5’ – CAC = H GUG – 5’5’ – CAA = Q GUU – 5’5’ – CAG = Q GUC – 5’
8CG5’ – CGU = R GCA – 5’5’ – CGC = R GCG – 5’5’ – CGA = R GCU – 5’5’ – CGG = R GCC – 5’
9AU5’ – AUU = I UAA – 5’5’ – AUC = I UAG – 5’5’ – AUA = I UAU – 5’5’ – AUG = M UAC – 5’
10AC5’ – ACU = T UGA – 5’5’ – ACC = T UGG – 5’5’ – ACA = T UGU – 5’5’ – ACG = T UGC – 5’
11AA5’ – AAU = N UUA – 5’5’ – AAC = N UUG – 5’5’ – AAA = K UUU – 5’5’ – AAG = K UUC – 5’
12AG5’ – AGU = S UCA – 5’5’ – AGC = S UCG – 5’5’ – AGA = R UCU – 5’5’ – AGG = R UCC – 5’
13GU5’ – GUU = V CAA – 5’5’ – GUC = V CAG – 5’5’ – GUA = V CAU – 5’5’ – GUG = V CAC – 5’
14GC5’ – GCU = A CGA – 5’5’ – GCC = A CGG – 5’5’ – GCA = A CGU – 5’5’ – GCG = A CGC – 5’
15GA5’ – GAU = D CUA – 5’5’ – GAC = D CUG – 5’5’ – GAA = E CUU – 5’5’ – GAG = E CUC – 5’
16GG5’ – GGU = G CCA – 5’5’ – GGC = G CCG – 5’5’ – GGA = G CCU – 5’5’ – GGG = G CCC – 5’

Table 1: Nirenberg’s first 16x4=64 codon/anticodon table (shown here for the first time in its complete form), being the codons with their anti-sense complements reordered according to the now-classical sequence of U, C, A, G [4,5], from top to bottom, and from left to right; in bold italics their corresponding amino acids; also shown, the direction of the start, or 5’– from left-to-right for the codons that codify in the upper lines, and from –5’, from right-to-left for the non-coding codons in the lower lines below the previous ones; this table has been modified from [3]: Here, the columns of the x axis have been numbered.

Efforts have been done to represent the genetic code in a binary way independent of the I Ching [13]; however, the I Ching is useful for the study the genetic code and was previously compared to a detailed treatment of Nirenberg’s Table in order to obtain quasi-symmetrical representations of chromosomes [10]. Likewise, in a similar fashion that the currently known letters of the genetic code are represented by AGTC, the first Hebrew mention of the genetic code3 uses four letters (grouped in 2×2), exactly as it happens in real life: being the four letters of the genetic code grouped by threes to produce 4×4×4=64 codons or words of exactly the same length per codon (three letters).

Now, the purpose of a teacher of the genetic code is to make it known by illustrating its extremely intelligent design and its complexities in the simplest way as possible; however, most of the current text books spend only one page representing the genetic code via Crick’s ‘square’ representation [4]. The word ‘genetic’ can be defined as: ‘the biological information that specifies the characteristics and metabolism of an organism’. The information in most of the biological life on earth is contained inside the nowfamous double helix [14] that swirls like in a ‘fusion dance’, tightly packaged within the chromosomes inside the nucleus of each living cell, except for the red blood cells that, by design, expel their nucleus, transforming themselves into biological ‘automatons’, or ‘robots’, for the cellular exchange between oxygen and CO2.

The genetic information that is transcribed from DNA to RNA in the nucleus is translated into proteins in the cellular cytoplasm, ending forming, these proteins, the structures and metabolism of our bodies.

The basic units that contain the information for our proteins are called ‘genes’, with each gene being composed of multiple nucleotides. The modular structure of the genes integrated by exons depicts the potential of a gene to produce several proteins through the ‘cut-andpaste’ (splicing) mechanism.

The words of the genetic code are the groups of three nucleotides called codons, producing the proteins that are formed by different combinations of the 20 amino acids, with some of those combinations being so frequent that these popular groups of amino acids, or of peptides, are considered as ‘domains’.

Next, the reverse-engineered Figures 1 and 2 are the square and the circular representations, respectively, of a functional tetrahedron discovered earlier [11] as a representation of the genetic code.

The periodicities or rotational correspondences of these representations of the genetic code are clearly identifiable when we rotate the quadrants by 90 degrees, keeping a positional and prominent correspondence for most of the essential hydrophobic amino acids located at the centre of the square of Figure 1 [12], remaining at the same relative rotational position in the circle shown in Figure 2 [8,9].

Also, and in order to detail the representation of the genetic code introduced in [10], the folded and ‘compressed’ 3-D representation of a double tetrahedron can be seen in Figures 3 and 4: being one functional and the other a 100% symmetrical geometry (with its Stella Octangula net shown in [10]).

The double genetic code was ‘physically’ compressed as represented in Figures 3 and 4, indicating that through the 3-D geometry of a Stella Octangula it’s possible to distinguish between the start amino acid Methionine, and the non-start Methionine [10].

A complex aspect of the genetic code thus far explored is the binary correlation between the three Cartesian combinations of nucleotides (hydrogen bonds, rings, and tautomerism) by using the millenary classical or original table of the I Ching [10], shown here compressed by groups in Figures 5 and 6.

We saw in the introduction the folded Stella Octangula, as well as its inner geometries derived from the computational analogy of the file compression or zipping; however, we are interested here in its expansion through the binary I Ching representation, to obtain, not only a cubical macro representation of the genetic code, integrated by 64×6=384 codons, but also its corresponding Sphered-Cube, a resemblance of the Cubed-Sphere widely used in Mathematics, Climatology, Geology, and Astronomy. The practical purpose of this is two-fold: First, to provide educational mnemonic devices; and second, to provide a useful core for molecular modeling in bioinformatics through a, not only highly informative, but also visually aesthetic bioinformatics tool to compare sequences of men and/or of other organisms; i.e., for the early detection of diseases, and/or to discover genomic compatibilities between related varieties of organisms currently misclassified as belonging to different species, or even to a different genus, instead of what they are in reality: varieties or sub-species, and vice versa. The geometries and binary structures used here are:

1) The net or pattern to obtain a folded Stella Octangula: http://web.archive.org/web/20101124045835/ http://korthalsaltes.com/model.php?name_en=stella%20octangula, and its hidden, internal Octahedron: http://liveweb.archive.org/http://mathworld.wolfram.com/Octahedron.html;

2) The original I Ching representation attributed to Fu-Xi: http://liveweb.archive.org/ http://www.laetusinpraesens.org/musings/images/unknow_files/fuxi.jpg,

3) The inner cubic 3-D space for Figure 22, obtained by using the freely available software located at: http://web.archive.org/web/20120625122500/ http://www.doka.ch/Excel3Dscatterplot.htm

Figures 25 and 26 show a practical application of the genetic code’s I Ching by using both the GenBank and BlastP to compare expressed sequences and a promoter between humans and a Neanderthal, as described there.

The next figures will be devoted to show the file compression and file expansion of the genetic code while advancing towards its macro representation. Figures 7 and 8 show the ‘file compression’, or zipping (zip), of codons according to their corresponding codified amino acids or functions.

If we now take the same groups shown in Figure 7 while removing their colours as a previous step before exhibiting their arrows that will be displaying the genetic code’s Yin/Yang directionality, we obtain the Figure 8, with its numbers and letters used here as a reference for the next step.

If we now demonstrate the directionality between the groups of codons according to their resulting amino acids or functions that are shown in Figure 8, we obtain the Yin/Yang arrows of Figure 9.

Figure 9 shows the directionality of the four basic components of the Yin/Yang as derived by the ancestral I Ching: The external arrows (flowing in opposite directions), being represented long ago by the ‘Old Yin’ and by the ‘Old Yang’ located at its extremes, while their younger and alternating pairing corresponds to the ‘Young Yang’ and to the ‘Young Yin’, respectively, being located at the centre while also moving in the opposite directions. These, in general, are also representing, both the process of DNA auto-replication (being the old strands of the double helix of DNA in the external lines, while the new strands are the internal), but also representing the process of DNA transcription into smaller segments of RNA.

If we now add the third Cartesian correlation, represented as well by the original symbols of the I Ching, we obtain what we see in Figure 10; and, if we go one step further and we add the arrows previously seen looking at two of its contiguous walls while standing on its third side, so we obtain Figure 11.

The potential for the file compression within the genetic code that we see in the preceding images can be further explored, showing here some of its numerous possibilities: In Figure 12 we see its zipped configuration in a circle while in Figure 13 we see such compression both in a tetrahedron and in a cube; these images are only some examples of the file compression applied to the genetic code analysis, which will potentially increase the speed of the analysis of sequences of DNA, RNA, or of proteins. If we now move one step further, we will be able to represent every possible binary combination of the genetic code in the classic I Ching (being only six possibilities available) within a cube, indicating that in addition to the three Cartesian correlated representations of the genetic code (in Appendixes. A we can see their common points), their three reciprocals are also included (see Appendixes. B); next, we will see the macro-representation of the genetic code in 6X as derived from the I Ching, either by codons (Figure 14), by amino acids (Figure 15), or by those colored codon groups that were shown before, but with no letters (Figure 16).

Increasing the level of complexity for our mentioned point of view as if standing inside the resulting cube, looking now to its two contiguous walls, and to its floor as if these were the three standard or normal correlations within the context shown by Figures 5, 6 and 10, we have the resulting Figure 17 for the codons, and Figure 18 for their resulting amino acids.

If we now look to the reciprocals of the previously defined as standard Cartesian I Ching genetic code correlations, we obtain Figures 19 and 20, and Appendix B. In Appendix C we see the coordinates used to obtain those standard three correlations shown in Figures 17 and 18, as well as their respective reciprocals shown in Figures 19 and 20, within the pair of 3-D vector Cartesian graphics (axis x, y, z) contained in the cube. The final product of this article will be a Sphered Cube for the genetic code (Figures 21 and 22).

If we now correlate the relative direction of the groups of codons according to their amino acids and/or functions, we obtain Figure 21 (for the standard correlation), and Figure 22 (for both the standard and the reciprocal correlations); and again, for these 3-D spatial versions were used, both the three standard correlations (Figure 21), and the reciprocals (Figure 22), of the Yin/Yang arrows shown in Figures 9 and 11.

Next, we are going to perform the ‘sphering’ of our genetic code’s macro-cube to obtain our first meaningful spherical representation of it, both by codons (Figure 23), and amino acids (Figure 24).

Appendixes D show the net without words necessary to obtain the 8×8×6=384 genetic code’s cube.

When we obtained the Sphered Cube by transforming our genetic code’s macro-cube into a sphere, we were surprised to realize that its resulting product was similar to the inverse reasoning of the ‘cubedsphere’, used initially in climatology for the subdivision of the earth [15-17], then in mechanics [18], in computational astrophysics [19,20], in mathematics [21], physics, chemistry, hydrology [22,23], climatology, geology, meteorology [24], etc.; however, this article seems to be the first time that such a structure has been originated through an opposite reasoning: the ‘Sphering’ of the cube instead of the ‘Cubing’ of a sphere, being here applied to the genetic code and to biology in general; an early model was built to represent our planet by using a smaller, 6×6=36 grids times six [15], and a similar image to the one obtained here (8×8=64), but by using an inverse reasoning, was obtained by [24, in its p. 840].

The keen observations done earlier by shCherbak, who has worked several aspects of the mathematical balance of the genetic code, are that: “…one may assume that natural computing can exist as well… such a computing could be essence of exact gene processing and scrambling… If that is the case, some cell organelles should work as biocomputers… the genetic code is connected more closely to abstract notions of arithmetic than with notions of physics or chemistry” [25]. Through the findings presented in this article, these statements of shCherbak have been corroborated.

And, in a similar way to what we have seen in our introduction when presenting the word ‘code’, shCherbak earlier equated the stop codons with zero (Arabic, Mayan, and Computational), deeming the zero as the supreme arithmetical abstraction and concluding that its use by any alphabet, including the genetic code, is an indicator of what he called ‘artificiality’ (being such artificiality in this case, the intelligent design and purpose of the genetic code: to bring and to sustain life), while affirming that the decimalization of the genetic code could be a special case of the general computational power of genomics and its molecular machinery, observing that the main reason for the origin of a numerical system is to do mathematical calculations; shCherbak concluded that: “there is no way to write or read any gene when no code is available” [26].4

As we mentioned earlier, the Cube and its Sphered Cube presented in this article can be used to compare sequences. Prompted by a reviewer and to demonstrate a practical application of our discovery, we will use here a compressed way of it by grouping its codons according to its resulting amino acids (Figure 25), where the matching sequenced genes can simultaneously run gene by gene within two cubes that belong to different organisms (in our example, Humans represented by the black arrows and Neanderthal by the red arrows), while pointing out to their differences by a resulting drawing.

More recently, while I was ready to resubmit this article, it was found that the patterns of expression are also very different between humans and neanderthals, highly differing in their methylation and deamination, a finding that may help explain why the expressed sequences of humans (the human amino acids) were extremely different (as shown in Figure25), even corresponding, perhaps, to totally different proteins, when compared to neanderthals (the Neanderthal’s amino acids). As an exercise, Figure 26 shows the differences on methylation/deamination between humans and ‘a proxy’ Neanderthal [36].

And, due to the fact that we are dealing with ancient sources of information, such as the I Ching and the Neanderthal’s genome, in Appendix E I dare to present what ancient manuscripts in Hebrew (shown in the introduction), Aramaic and Greek, that are contained within the Bible have to say in relation to it. Otherwise, in order to analyze more in detail the currently full Neanderthal and Human sequences freely available through the Max Planck institute5, I will require of a greater computer power and memory space, as well as the development of a computational program with the Sphered Cube as its engine; for example, I performed earlier an unpublished analysis of some of the differences in the mitochondrial DNA between Humans and Neanderthal,6 however, the current study will be more graphically appealing.

Elsewhere, an example of a human polymorphism [9], and the comparison of the human genome with other organisms has been presented [9,37], as well as one molecular application of the genetic code in 3-D to differentiate the start from non-start Methionines within proteins [10].

Also, and apart of exploring the 3-D properties of the genetic code, cell cultures in 3-D have been recently used by the author of this article in an attempt to simulate the current conditions of living tissues through the modeling of the origin and attenuation of atherosclerosis [38] in human osteogenic trans-differentiating vascular smooth muscle cells (VSMCs). We saw that Lyso-phosphatidylcholine (LPC) at physiological concentrations (10 nm), instead of the entire Low Density Lipoprotein (LDL) or ‘bad cholesterol’, seemed to have been the main responsible for the development of calcification in atherosclerosis [39-41], being this a defensive response of the arterial muscular cells against the detergent-like effects of the LPC, a mechanism for those cells to protect themselves.

The optimization of the 3-D cell culture methodology for atherosclerosis required more than two years; however, reference [42], not only added the LPC that was successfully demonstrated by us [39], while using my methodologies and materials with no credits, but also ruined our experimental model by adding two extra, non-necessary and non-physiological reagents: β-glycerophosphate (7.5 mM) and ascorbic acid (50 μg/ml), being these additions reflected in their work by the retention of hydroxyapatite within the VSMCs of their 3-D cluster (seen at the end of their third composite illustration), instead of being normally secreted, as it was shown in all of our Figures [38].

Our original work [38] demonstrates that a 3-D perspective is highly useful in biomedical research, not only for its 3-D cell cultures, but also for the current study of the genetic code in 3-D as shown here.

Tidjani Negadi, a dedicated researcher of the genetic code, has been able to find numerous instances of the number 384 within the genetic code, a number here obtained through our I Ching cube (64×6=384); Negadi presents at least 15 calculations to obtain the number 384 within the genetic code:

1) 360+tau (360)=360+24=384 [43] (number equal to the number of atoms in the 20 amino acids, side-chain and block [44], as well as the total number of nucleotides in the 64 RNA-codons and in the 64 DNA codons [45]),

2) (73+11) +300=84+300=384 [43],

3) (99+74+7)+(73+41+11+20+35)+σ(360)=180+204=384 [43],

4) 360+(5+5+5)+9=384 [43],

5) 2x192=384 [43,45],

6) (3412/2x)-1255-67=384 (from the irregular L- and D- (integer) tetrahedron, “a warehouse of biological information”) [43],

7) 192+192=384 [45] (The mean of the four pairs equal 192, the total number of nucleobases in the 64 codons, but also the total number of codons in DNA and RNA (2×64) [46]),

8) 768/2=384 [46],

9) phi(2)(psi)=384 [46],

10) psi upper bar(0,1,0,0)(psi)=psi upper bar(0,0,3,0)(psi)=384 [46],

11) Writing 3276+676 and introducing the identity “-384+384” (384=384 or “384-384=0”), we get 2892+1060 which is the “partition of nucleon-numbers between the four-codon set and the non-four codon set” [46],

12) phi(1152)=384 [46],

13) 313+71=384=204+(9x20),

14) Filatov [47] also “established a nucleon number balance in two pairs of faces 628+627=626+629=1255. Now computing the atom number in the amino acids in the two pairs of faces, we find 108+97=205 and 104+99=203 (without the side-chains) or 198+187=385 and 194+189=383 (including the side-chains). These are near balances and the average of the two pairs gives 204 and 384 atoms, respectively, for the correct atom number. Now from our numbers 139 (34th prime) and 1304, we could take the sum of the prime factors and the prime indices to get 173 and 210 with sum 383. This is the nucleon number in one of the two pairs of faces mentioned above …and we could reach 384 by just adding the Omega-function of the prime number 139 which is simply 1. Also, by re-arranging the terms we have (139+34+1+6)=180 and (163+38+3)=204. The first number is the total atom number in the 20 blocks and second is the atom number in the 20 side-chains”. So, concluding the last computation, we have (139+34+1+6)+(163+38+3 )=384=180+204 [The points 13 and 14 were found in Negadi’s blog].

15) Plus, a music related one: “The numbers 192 and 384 seem very important numbers” [Plato did choose 192 as a starting point while Timaeus the Locrian did choose 384] “…they are basic numbers in the Pythagorean music tuning system” [45] (also see [44]).

Figure 27 shows how both our 384-grid Cube and Sphered Cube representation of the genetic code can be compared to the Pythagorean music series.

Using only proportions of 2 and 3 (± mean by 1/3, in row F), the frequency relationships seen in Figure 27, when applied to the building-up of the I Ching genetic code are: 64, 128, 192, 256, and 384; however, the I Ching’s successive number 320 that appears between 256 and 384 is not present; the numbers that are actually present, in the current context are: 64×1=64, 64×2=128, 64×3=192, 64×4=256, [the next consecutive one, 64×5=320, is absent in the Pythagorean music system], and 64×6=384. Additional multiples of 64 that are seen in the Pythagorean musical scale are: 64×8=512, 64×9=576 and 64×12=768.

Notably, if the numbers within the left side column (headed by 11- in the row F) are multiplied by 10, all of them became also multiples of 64, including the previously missing 64×5=320, plus 64×15=960, 64×45=2880, and 64×135=8640.

Regarding such discrepancy seen in lines C and E (and even B) of the Pythagorean music series, Ray Tomes observes: “C is incorrectly related to E and B… as Galilei observed, C and E want to be in the proportion 4:5 (which is 64:80) and we have 64:81 so it just misses… The equitempered scale is designed around the ratio 2 because each semitone is a ratio of 1 to the twelfth root of 2. As it turns out this accommodates ratios of 3 almost perfectly, which is of course why it was chosen… I invented a system which I call AJI for Automatic Just Intonation. I realised [sic] that an electronic keyboard doesn’t have to make compromises when selecting frequencies as there is no reason that a single key cannot produce different frequencies depending on the circumstances…” In our case, as mentioned earlier, we multiplied by ten the whole column 11- to make it ‘fit’ the pattern of our design, and it worked, providing us with an artificially ‘perfect’ symmetry, as seen in Figure 27.

16) In our case, the number 384 was also obtained by counting the individual binary components within any of the I Ching’s genetic code tables (Figures 5and 10), where we see that each cell is composed by a trigram from the y axis, over which a trigram from the x axis is added, integrating an hexagram, so when multiplying one hexagram by the totality of the cells, we also obtain the number 384: 6×64=384.

Finally, and as the last project of my free programming edX Harvard University class (cs50x), I decided to put motion to my earlier work to illustrate the complementary pairing by rows of the basic structure of the Figure 5 of this article, which is similar, but not identical, to the first Table (T.1) of Figure 1 of my previously related publication [10]. The project in its final form can be seen at: http://www.youtube.com/watch?v=0aacCShsmWo, coupled to an explanation of it. The basic 8 sprites and 9 scripts used to develop the animation for this video can be downloaded at its M.I.T. ‘source code’:

http://scratch.mit.edu/projects/fdocc/3226378 Our current experimental lab work dealing with 3-D cubic cylinders and crystallized geometries can also be seen at: http://youtu.be/oYd96rj_51Q

Biodiversity on earth is based on the flexibility and plasticity of the genetic code. This is just the start of studies demonstrating the enormous plasticity of the genetic code, being its current representations and its future possible ones like a metaphorical reflection of the real plasticity contained within living organisms [10], being this the reason for their great variability and adaptation as seen in the exuberant biodiversity that surrounds us.

We also showed here the reverse engineering in 2-D of a previously obtained functional tetrahedron in 3-D [10]. The resulting products were a square (Figure 1) and a circle (Figure 2). We also folded a 3-D double genetic code representation composed by a Stella Octangula (Figures 3 and 4) found earlier [10].

The rotation seen in Figure 9 from row 1 to row 4 can also be perceived as evocative of the actual helical rotation that is seen in the double helix [14]. The two circular and compressed representations of the genetic code with the stop codon at the centre that are shown in Figure 12, if they were rotating through a pin at their centre, while an external arrow were pointing at their respective codons, if it were stopped at a line between two of these portions, it will go then to the stop codons located at its centre.

The inclined rotational axis of 45 degrees shown in Figures 14-16 (represented there by the intermittent circles, their net can be seen in Appendix D), remind us of the inclined orbital axis of the earth; these diagonals of computational geometry within a cube are deemed to be located either as a π/4 or a 3π/4, depending if the relative view is from left to right or vice versa [48-50] shows the software used to develop (Figure 22).

The zipping or compression of the codons by groups, both in the 3-D and in 2-D representations of the genetic code discussed here may increase the speed of our comparison of sequences, as well as our explorations of the genetic code within curved spaces, spaces studied in my classes related to the foundations of computer graphics;7 i.e., the resulting cubed-sphere may be useful to compare sequences, either from wildlife or from humans in an appealing visually graphic way, noticing also that the file expansion shown here is not the same as the unzipping of the computer files, being this the product of the restoration to the same size of the original file as it was before, while the file expansion shown here results in the increase of informational content.

Here it was also shown that human genes translated from the brain and from a promoter in osteoblasts, are compared to their corresponding Neanderthal sequences to show their differences and to demonstrate the use of the I Ching’s genetic code in 3-D (Figures 25 and 26). After these findings, we can declare with confidence that the oldest known representation of the Genetic Code is the binary I Ching, also known as the Book of Changes or Book of Mutations, corresponding these changes to biological changes or mutations, a book preserved by the Chinese, but evidently being older than them, a survivor of the burning of ancient documents by the Emperor Chi (also known as Qi).

In relation to this and to my previous works, a recent article declared: “Those two properties (ie, symmetry and periodicity) act as the harmony between the chosen geometry and the biological reality. Graphical representation of DNA sequences based on mono, di, trinucleotides, etc. need to consider this harmony. Otherwise, it would merely be an instance of displaying the nucleotides (eg, mononucleotide, dinucleotide, codon) which have little biological sense” [51]; with the current work, the corroboration of such observations is advanced.

A recent article [52], making also reference to one of my earliest works related on the 2-D circular genetic code, when alluding to “the palindromes of the genetic code”, declared: “that some researchers have tried to restrict at peculiar symmetries”; my answer is that in order to learn from the continuous wisdom manifested within the genetic code, certainly the less limited our analyses are, the best their output will be; I strive to encourage every student and researcher of the genetic code to explore the fullness of its possibilities without dismay.

Once more, the enormous plasticity that we see in the study of the genetic code is a reflection of the real plasticity contained within living organisms, being this the ultimate reason for their great variability and adaptation. The great variability seen in nature can better be understood by studying its source, the genetic code; for example, in my earlier studies [8,9] of the 2-D classic circular genetic code [5], I recently realized that the Hydroxyl amino acids, those ones that are phosphorilatable in order to activate the dormant enzymes, are also located in a balancing way within quadrant 1 (4 Ser+2 Tyr), and quadrant 3 (2 Ser+4 Thr).

The use of the Yin/Yang (Yin and Yang) in this article is within the context of the Genetic Code in 3-D, something apparently known and fully represented since a long time; i.e., in the I Ching or Book of Changes; however, the use of this concept is not new within medicine and molecular biology; a few set of examples from the numerous found within PubMed (currently ~500 references), are given at [53-85].

Here, by grouping the reiterations or “redundancies” of the genetic code, reiterations apparently designed to preserve its functionality while allowing the adaptability of living organisms, a more detailed picture appeared, both in 2-D and in 3-D, when comparing the Cartesian binary combinations of the properties of nucleotides, then the directions of the Yin and of the Yang arrows suddenly appeared. It is precisely this reiterated redundancy of the genetic code, the basis for its ‘file compression’ in bioinformatics and biodiversity; additionally, it was shown that the number 384, obtained in this article by multiplying 64x6 within all the possible binary comparisons of the genetic code derived from the I Ching, has also been found within the mathematics derived from the molecules that integrate the genetic code, as seen above when our current work is compared with that of Negadi [43,45-46], and others [44,47].

On concluding, I may say that the main findings of this work are: That the Yin/Yang’s genetic code directions are similar to those that are presented by the actual separate DNA strands in relation to their autoreplicating new strands, and/or in relation to their new RNA products, specially by the plus strand at the moment of transcription for the last example. Also including the minus strand, and its smaller replicating segments known as the Okazaki fragments for the DNA autoreplication. The findings presented here can help in bioinformatics; i.e., by locating at the center of a computer program the genetic code in its cubic or spherical form, not only to compare sequences in an attractive or aesthetic way, but also to provide didactic illustrations of these proceedings.

I thank God for giving me the time and strength to do this work, and to my wife Tracy L. Duncan for her unconditional support; to Reid and Sheryl McNally, Stephen and Brenda Sparks, Cathy Keener, Eliezer Mendelssohn, Fernando Paularena, Glenn Lewis, Lynn D. Wills, David J. Lewis, Margina Huq, Sheldon Yoder, Bruno Bazan, Zachary Greenbauer, Chris Caldwell, Daniel Garcia, Mark Skipski, Maria Lantsman, Olivia Nguyen, and to the rest of my coworkers that make all the daily work enjoyable; and also to my parents, sisters, nephews, to the inlaws, and to my students Ana Baleva and Karen Scarbrough, as well as to my spiritual and physical families and friends. This manuscript was supported in part by the NIH grant T32 HL-07812.

1http://web.archive.org/web/20101129051637/http://rumkin.com/tools/cipher/ numbers.php

2 Abbr.: Amino acids: A: alanine (Ala), V: valine (Val), I: isoleucine (Ile), L: leucine (Leu), M: methionine (Met), F: phenylalanine (Phe), W: tryptophan (Trp), D: aspartic acid (Asp), N: asparagine (Asn), E: glutamic acid (Glu), Q: glutamine (Gln), R: arginine (Arg), K: lysine (Lys), S: serine (Ser), T: threonine (Thr), G: glycine (Gly), P: proline (Pro), H: histidine (His), C: cysteine (Cys), Y: tyrosine (Tyr). Nucleotides (nt): U: uracil, C: cytosine, A: adenine, G: guanine, T: thymine. n: any nt, r: purines, y: pyrimidines.

3 Psalm 139:16, http://biblehub.com/interlinear/psalms/139-16.htm.

4 Its mdi file is available at: http://liveweb.archive.org/http://tinyurl.com/shcherbak

5http://cdna.eva.mpg.de/neandertal

6http://fdocc.ucoz.com/index/0-121

7 Class CS184.1x Foundations of Computer Graphics, imparted by Ramamoorthi, from U.C. Berkeley (free at edX).

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