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+---
+sort: 5
+spin: 1
+span: replicate
+suit: intro
+---
+# [Triangular waves](https://gist.github.com/eq19/e9832026b5b78f694e4ad22c3eb6c3ef#bonding-position)
+
+>A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd harmonics. However, ***the higher harmonics roll off much faster*** than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse) _([Wikipedia](https://en.wikipedia.org/wiki/Triangle_wave))_.
+
+[![Making Triangular Waves](https://user-images.githubusercontent.com/8466209/225824209-ba2b9fe0-1a29-4208-940e-3351243ab0ba.png)](https://www.primesdemystified.com/First1000Primes.html)
+
+Hamilton's program for proving the Poincaré conjecture involves first putting a Riemannian metric on the unknown simply connected closed 3-manifold. Such mechanism is discussed in detail by _[16th prime identity](https://gist.github.com/eq19/dc30497160f3389546d177da901537d9)_.
+
+![](https://user-images.githubusercontent.com/8466209/219422862-38fd1aae-0ae7-4ba3-8e88-db10a2cf522f.png)
+
+There are 243 prime pairs among the first 1000 primes that sum to 7920, where 243 x 7920 gives additive sum 1,924,560 (and we note that 7920 = 22 x 360 = 2(1092 − 892), which equates to 1,924,560/(2(1092 − 892) = 243.).
+
+![default](https://user-images.githubusercontent.com/8466209/225643958-7933e50d-6cd3-42b4-8de1-eca70687e713.png)
+
+Consider these two (2) numbers are laid side by side as part of Fibonacci sequence on the [Pascal Triangle](https://gist.github.com/eq19/b32915925d9d365e2e9351f0c4ed786e#file-bilateral-md). Thus the above is actually represesenting the pattern of _[true-prime-pairs](https://gist.github.com/eq19/0ce5848f7ad62dc46dedfaa430069857#true-prime-pairs)_ via _[bilateral 9 sums](https://gist.github.com/eq19/b32915925d9d365e2e9351f0c4ed786e#file-bilateral-md)_.
+
+***63 = 21 x 3 = 7 x 3 x 3***
+
+[![Fibonacci Sequence Numbers](https://user-images.githubusercontent.com/8466209/218188579-8b7abe41-3137-41fd-8620-c141a719b493.png)](https://gist.github.com/eq19/b32915925d9d365e2e9351f0c4ed786e#file-bilateral-md)
+
+Meanwhile the sum of ***28 and 35***, which is 63, has a digital root of 6+3 = 9 and is landed precisely on the _[15's cell](https://gist.github.com/eq19/b32915925d9d365e2e9351f0c4ed786e#file-default-md)_ of the prime hexagon. So the interaction with the [seven (7) objects](https://gist.github.com/eq19/b32915925d9d365e2e9351f0c4ed786e#file-bilateral-md) is made through the _[object of 36](https://gist.github.com/eq19/b32915925d9d365e2e9351f0c4ed786e#file-default-md)_ which has the same digital root of 9.
+
+>la position du début de l'emplacement, dans les décimales du nombre e, de la séquence ***0719425863***, qui est la première occurrence d'une séquence de longueur ***10 contenant*** chaque chiffre une et une seule fois _([Wikipedia](https://fr.wikipedia.org/wiki/1_729_(nombre)))_.
+
+![](https://user-images.githubusercontent.com/36441664/73480257-261c4e00-43cc-11ea-8c80-b2cd7b8d98b0.png)
+
+This _[1729 objects](https://gist.github.com/eq19/e9832026b5b78f694e4ad22c3eb6c3ef#prime-recycling)_ is passing the first cycle of _[True Prime Pairs](https://www.eq19.com/addition/file02.html#true-prime-pairs)_ so it will be located by the starting point of 286 objects of 2nd identity. This balancing will exchange ***786 and 500 objects in pair***.
+
+***1729 = 786 + 157 + 786 = 500 + 286 + 157 + 286 + 500***
+
+[![Euleur Gap](https://user-images.githubusercontent.com/8466209/220001571-bddc7fee-5753-4ee3-b145-b3b1bb49a354.png)](https://gist.github.com/eq19/e9832026b5b78f694e4ad22c3eb6c3ef#prime-identity)
+
+The basic idea is to try to "improve" this metric; for example, if the metric can be improved enough so that it has constant positive curvature, then according to classical results in Riemannian geometry, it must be the 3-sphere. 
+
+![](https://user-images.githubusercontent.com/8466209/195509190-6926e3cd-807a-44ef-8549-b0fc1f481a2d.png)
+
+Hamilton prescribed the "[Ricci flow equations](https://gist.github.com/eq19/e9832026b5b78f694e4ad22c3eb6c3ef#ricci-flow)" for improving the metric. Ricci flow expands the negative curvature part of the manifold and contracts the positive curvature part. In some cases, Hamilton was able to show that this works
+
+## System Images
+
+![](https://user-images.githubusercontent.com/36441664/125712858-fd635048-1970-48f7-a063-758c9c99a3a7.png)
+
+[![](https://user-images.githubusercontent.com/8466209/209758101-c62460d5-0fb0-461a-ac64-931e07bd0828.png)](https://user-images.githubusercontent.com/8466209/205960400-3061d93a-b3cc-4cd6-84e6-359a9314869d.png)
+
+Pass a startup script from Cloud Storage for the scripts greater than 256 KB. 
+
+[![Startup script](https://user-images.githubusercontent.com/8466209/221899049-3a88854a-1836-4960-832c-a2f849f25f68.png)](https://cloud.google.com/compute/docs/instances/startup-scripts)
+
+**Artifact Registry** expands on the capabilities of Container Registry and is the recommended container registry for Google Cloud. 
+
+![2023-03-01](https://user-images.githubusercontent.com/8466209/222013722-6c410230-2665-4636-9071-9d2453ae5758.png)
+
+A [Cloud Build trigger](https://cloud.google.com/build/docs/automating-builds/create-manage-triggers) automatically starts a build whenever you make any changes to your source code. You can configure the trigger to build your code on any changes to the source repository or only changes that match certain criteria.
+
+[![](https://user-images.githubusercontent.com/8466209/222014481-8af35f36-f6cf-4260-9e47-f22f5017ed12.png)](https://cloud.google.com/build/docs/automating-builds/create-manage-triggers)
+
+[![8d0zCSMHxpgkFZTmXCGk](https://user-images.githubusercontent.com/8466209/210665303-b56792d5-8442-4e8a-a1fb-ae95ae6235d5.png)
+](https://gist.github.com/eq19/c9bdc2bbe55f2d162535023c8d321831)
+
+Compute Engine supplies an up-to-date [Container-Optimized OS (COS)](https://cloud.google.com/container-optimized-os/docs) image with Docker installed and launches your container when your VM starts. See below the [access control](https://cloud.google.com/artifact-registry/docs/access-control) for information about private registry permissions.
+
+> DNA is read by DNA polymerase in the 3′ to 5′ direction, meaning the new strand is synthesized in the 5' to 3' direction. Since the leading and lagging strand templates are oriented in opposite directions at the replication fork, *****a major issue***** is how to achieve synthesis of new lagging strand DNA, whose direction of synthesis is opposite to the direction of the growing replication fork __(Source: [Wikipedia](https://en.wikipedia.org/wiki/DNA_replication#Replication_for))__.
+
+[![default](https://user-images.githubusercontent.com/8466209/199140400-d85fef37-1c7d-455b-bb56-6841694ed648.png)](https://en.wikipedia.org/wiki/DNA_replication#Replication_fork)
+
+Means this field remains unthoucheable. Up to the previous topic of [quantum](https://gist.github.com/eq19/8cab5e72d52ecb338a2f2187082a1699) we brought you [AI](https://en.wikipedia.org/wiki/Artificial_intelligence). A touchable intelligence. By this project we would like to introduce a brand new intelligence called **UI** which is the abreviation of [Untouched Inteligence](https://github.com/chetabahana/tensorflow/wiki/Untouched-Intelligences)
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+
+![cubes](https://user-images.githubusercontent.com/8466209/205804217-313e8f74-56d1-4401-be92-e3d3c95da055.jpg)
+
+![prime](https://user-images.githubusercontent.com/36441664/87958552-dea18f80-cadb-11ea-9499-6c2ee580a5ca.png)
+
+![ulam](https://upload.wikimedia.org/wikipedia/commons/2/25/Hexgrid_prime_number_spiral.svg)
+[![default](https://user-images.githubusercontent.com/8466209/205531184-66086f89-ee4b-4536-9e22-9b1dc946b21e.png)](https://gist.github.com/eq19/f78d4470250720fb18111165564d555f#file-default-md)
+
+![lexer](https://user-images.githubusercontent.com/8466209/204159538-6f87b60c-6a32-486d-aa55-c9af2db7eda6.png)
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+```liquid
+  -----------------------+----+----+----+----+----+----+----+----+----+-----
+   True Prime Pairs Δ    |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 | Sum 
+  =======================+====+====+====+====+====+====+====+====+====+=====
+   19 → π(10)            |  2 |  3 |  5 |  7 |  - |  - |  - |  - |  - | 4th  → base (30)
+  -----------------------+----+----+----+----+----+----+----+----+----+-----
+   17 → π(20)            | 11 | 13 | 17 | 19 |  - |  - |  - |  - |  - | 8th  → maps (31)
+  -----------------------+----+----+----+----+----+----+----+----+----+-----
+   13 → π(30) → 12 (Δ1)  | 23 | 29 |  - |  - |  - |  - |  - |  - |  - |10th  → feed (32)
+  =======================+====+====+====+====+====+====+====+====+====+===== 
+   11 → π(42)            | 31 | 37 | 41 |  - |  - |  - |  - |  - |  - |13th  → lexer (33)
+  -----------------------+----+----+----+----+----+----+----+----+----+----- 
+    7 → π(60) → 19 (Δ12) | 43 | 47 | 53 | 59 |  - |  - |  - |  - |  - |17th  → parser (34)
+  -----------------------+----+----+----+----+----+----+----+----+----+----- 
+    5 → π(72) → 18 (Δ13) | 61 | 67 | 71 |  - |  - |  - |  - |  - |  - |20th  → syntax (35)
+  =======================+====+====+====+====+====+====+====+====+====+===== 
+    3,2 → 18+13+12 → 43  | 73 | 79 | 83 | 89 | 97 | 101| 103| 107| 109|29th  → grammar (36)        
+  =======================+====+====+====+====+====+====+====+====+====+=====
+           Δ                                                            Δ
+  12+13+(18+18)+13+12   ← 36th-Δ1=151-1=150=100+2x(13+12)   ←   30th = 113 = 114-1
+  ```
+
+![2022-12-06 (2)](https://user-images.githubusercontent.com/8466209/205963196-7e1f4fc3-1833-4b64-9fb4-f79e6cd4637f.png)
+
+![2022-12-06 (3)](https://user-images.githubusercontent.com/8466209/205964436-c9636678-6679-48f7-938a-e307912600ef.png)
+
+![2022-12-06 (1)](https://user-images.githubusercontent.com/8466209/205947618-ef28a677-f961-4b78-9e79-e8a9ffe824c5.png)
+
+![2022-12-06 (1)](https://user-images.githubusercontent.com/8466209/205960400-3061d93a-b3cc-4cd6-84e6-359a9314869d.png)
diff --git a/exponentiation/span14/gist04.html b/exponentiation/span14/gist04.html
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+![2022-12-06 (5)](https://user-images.githubusercontent.com/8466209/205967762-69f6aaee-f1fc-4b26-a0cc-515257b6d9e7.png)
+
+![2022-12-06 (4)](https://user-images.githubusercontent.com/8466209/205967707-254fc836-72f4-45fc-9d0e-90c5e0bda465.png)
+
+![2022-12-10 (1)](https://user-images.githubusercontent.com/8466209/206849387-d530c690-c1ed-4496-9748-355f798d44fa.png)
diff --git a/exponentiation/span14/gist05.html b/exponentiation/span14/gist05.html
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+```liquid
+1st layer:
+It has a total of 1000 numbers
+Total primes = π(1000) = 168 primes
+
+2nd layer:
+It will start by π(168)+1 as the 40th prime
+It has 100x100 numbers or π(π(10000)) = 201 primes
+Total cum primes = 168 + (201-40) = 168+161 = 329 primes
+
+3rd layer:
+Behave the same as 2nd layer which has a total of 329 primes
+The primes will start by π(π(π(1000th prime)))+1 as the 40th prime
+This 1000 primes will become 1000 numbers by 1st layer of the next level
+Total of all primes = 329 + (329-40) = 329+289 = 618 = 619-1 = 619 primes - Δ1 
+```
+
+The 619 is the **114th prime**. It has the last index of 6 with the prime 19 where as **6x19** is also 114. When we multiply the 114 [minor hexagon](https://www.hexspin.com/minor-hexagons/) again by six (6) minus the duplicaton within those layers we got 684-30-30-5 = 684-65 which is also 619.
+
+[![metatron](https://user-images.githubusercontent.com/8466209/90985852-ca542500-e5a8-11ea-9027-9bfdcbe37966.jpg)](https://gist.github.com/eq19/0ce5848f7ad62dc46dedfaa430069857#file-1_prime-md)
+
+[![gannchart](https://user-images.githubusercontent.com/8466209/206868681-2cddf780-48f1-4dc5-9ef4-981b402a15d5.png)
+](https://gist.github.com/eq19/80c8098f16f3e6ca06893b17a02d910e#file-mapping-md)
+
+Despite the correlation between the bilateral 9 sums with the Fibonacci sequence, some people has discovered many things. The sequence of ***(55, 89, 144)*** shows an almost mystical power by its mysterious accuracy in nature and even _[in trading](https://www.tradingsim.com/day-trading/fibonacci-trading)_.
+
+[![image](https://user-images.githubusercontent.com/8466209/216902149-2d017e85-3d95-4484-bc6c-59a4b2079137.png)](https://www.tradingsim.com/day-trading/fibonacci-trading)
+
+![2022-12-10](https://user-images.githubusercontent.com/8466209/206884467-05a33d48-5cdc-48fd-9b75-5be406100f06.png)
+
+[![cycle](https://user-images.githubusercontent.com/8466209/205531184-66086f89-ee4b-4536-9e22-9b1dc946b21e.png)](https://gist.github.com/eq19/f78d4470250720fb18111165564d555f#file-default-md)
+
+![2022-12-14 (1)](https://user-images.githubusercontent.com/8466209/207418457-4db2048d-e003-492d-a78c-bc2ded725c74.png)
diff --git a/exponentiation/span14/gist06.html b/exponentiation/span14/gist06.html
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+![gotek](https://user-images.githubusercontent.com/36441664/123180870-57f8ed80-d4b6-11eb-89e1-cf8786d0dc14.jpg)
+
+![2022-12-11](https://user-images.githubusercontent.com/8466209/206888301-0ba40a6a-2caa-4dfe-92ef-5a9db590f85d.png)
+
+![io](https://user-images.githubusercontent.com/36441664/123179535-cd16f380-d4b3-11eb-95c3-0a5b7ef21ad4.jpg)
+
+![floppies](https://user-images.githubusercontent.com/36441664/123187010-84ffcd00-d4c3-11eb-8aad-6c40553b752b.png)
+
+[![2022-12-11 (1)](https://user-images.githubusercontent.com/8466209/206888720-159281ee-1f74-4684-aca3-963dd3c882ab.png)](https://docs.github.com/en/actions/learn-github-actions/usage-limits-billing-and-administration#usage-limits)
+
+![neural](https://user-images.githubusercontent.com/36441664/123424569-377e7f80-d5eb-11eb-9c85-1f5d72f14eed.png)
diff --git a/exponentiation/span14/gist07.html b/exponentiation/span14/gist07.html
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+```
+---+-----+-----
+ 1 | 1   | 4
+---+-----+-----
+ 2 |{5}  |{8}
+---+-----+-----
+ 3 | 9   |{17}
+---+-----+-----
+ 4 |{18} |{23}
+---+-----+-----
+ 5 | 24  | 27
+---+-----+-----
+ 6 | 28  | 34
+---+-----+-----
+ 7 |{35} |{41}
+---+-----+-----
+ 8 |{42} |{52}
+---+-----+-----
+```
+
+![](https://user-images.githubusercontent.com/36441664/85040788-5e71cc80-b1b3-11ea-962f-cf26ee3419e2.jpg)
+
+![](https://user-images.githubusercontent.com/36441664/84902333-e6ce6f80-b0d6-11ea-8289-aac5e1961cd6.gif)
+
+![equilateral](https://user-images.githubusercontent.com/8466209/205804972-0570afbc-3781-473c-bdc7-5e6927351de5.gif)
+
+![](https://user-images.githubusercontent.com/36441664/85527369-044e8c80-b635-11ea-8f32-25a6b813a9b1.gif)
+
+![](https://user-images.githubusercontent.com/36441664/85907724-ae4c3580-b83c-11ea-85c5-c9c20e13bd08.png)
+
+![](https://user-images.githubusercontent.com/36441664/85039712-32098080-b1b2-11ea-80f0-3e7e767a2ba1.jpg)
+
+>This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding. _([Clay Institute](https://www.claymath.org/millennium-problems))_.
+
+[![Navier–Stokes Equation](https://user-images.githubusercontent.com/8466209/224287999-37cd0692-bf51-4b99-bb45-8c49e5db58e3.png)](https://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation)
+
+By _[this paper](https://www.sciencedirect.com/science/article/pii/S0362546X23000111)_ it is derived a sufficient condition keeping energy conservation in terms of the gradient of the velocity for the weak solutions of the compressible Euler equations for both the non-vacuum and the vacuum cases.
diff --git a/exponentiation/span14/index.html b/exponentiation/span14/index.html
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+## Electron scattering
+
+Electron-electron scattering is described by one of the earliest published Feynman diagrams (featured in “Sightings,” September–October 2003). One electron (solid line at bottom right) shoots out a forcecarrying particle—a virtual photon (wavy line)—which then smacks into the second electron (solid line at bottom left). 
+
+>The first electron recoils backward, while the second electron gets pushed off its original course. The diagram thus sketches a quantum-mechanical view of how particles with the same charge repel each other. As suggested by the term “Space-Time Approach” in the title of the article that accompanied this diagram, Feynman originally drew diagrams in which the dimensions were space and time; here the horizontal axis represents space. 
+
+[![default](https://user-images.githubusercontent.com/8466209/224118806-b00758c2-0885-46c3-b574-566dffedf95f.png)](https://en.wikipedia.org/wiki/Feynman_diagram)
+
+![image](https://user-images.githubusercontent.com/8466209/240317298-4d42ea8d-ef95-4a33-9a6c-518149dbc02a.png)
+
+![image](https://user-images.githubusercontent.com/8466209/223643996-1ea5cb89-324e-446c-9c9f-8c94c39909b3.png)
+
+[![default](https://user-images.githubusercontent.com/8466209/224117996-fd3f0e3b-2ef6-43f5-961d-e90b616cbccc.png)](http://web.mit.edu/dikaiser/www/FdsAmSci.pdf)
+
+![](https://user-images.githubusercontent.com/8466209/222020262-cffc624a-b92c-4137-b76a-5423df7fc5ad.png)
+
+![Untitled](https://user-images.githubusercontent.com/8466209/207262180-7a0928a1-0c16-4193-9611-0626ad884ab1.png)
+
+![Untitled](https://user-images.githubusercontent.com/8466209/207363317-e8816b5d-c7b4-43e3-a120-0509641de4eb.png)
+
+![207363317-e8816b5d-c7b4-43e3-a120-0509641de4eb](https://user-images.githubusercontent.com/8466209/207490064-f96db6e0-33fe-4c71-9fd9-9f13e71410db.png)
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diff --git a/exponentiation/span15/gist04.html b/exponentiation/span15/gist04.html