GET https://rct.dev.bbntimes.com/technology/the-tao-of-tau-intersection-of-tau-complex-numbers-code-libraries-and-javascript-in-analytics

Response
200 OK
IP
216.73.216.240
Profiled on
Token
1ac37a

ArticleController :: show

Request

GET Parameters

None

POST Parameters

None

Uploaded Files

None

Request Attributes

Key Value
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"App\Controller\ArticleController::show"
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"security.firewall.map.context.main"
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    3720 => Symfony\Component\WebLink\Link {#3720
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"article_show"
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_security_firewall_run 
"_security_main"
_stopwatch_token 
"9f8e3e"
category 
"technology"
slug 
"the-tao-of-tau-intersection-of-tau-complex-numbers-code-libraries-and-javascript-in-analytics"

Request Headers

Header Value
accept 
"*/*"
accept-encoding 
"gzip, br, zstd, deflate"
connection 
"close"
cookie 
"PHPSESSID=vts6vkj800g6cuitjfcmnrirq3"
host 
"rct.dev.bbntimes.com"
user-agent 
"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)"
x-php-ob-level 
"1"

Request Content

Request content not available (it was retrieved as a resource).

Response

Response Headers

Header Value
cache-control 
"no-cache, private"
content-type 
"text/html; charset=UTF-8"
date 
"Sat, 21 Jun 2025 17:14:20 GMT"
link 
"</build/runtime.js>; rel="preload"; as="script",</build/644.js>; rel="preload"; as="script",</build/502.js>; rel="preload"; as="script",</build/app.js>; rel="preload"; as="script",</build/view-more.js>; rel="preload"; as="script",</build/term-condition.js>; rel="preload"; as="script",</build/contact.js>; rel="preload"; as="script",</build/scroll-infinite-article.js>; rel="preload"; as="script",</build/app.css>; rel="preload"; as="style",</build/cookie-style.css>; rel="preload"; as="style",</build/term-condition-css.css>; rel="preload"; as="style",</build/contact-css.css>; rel="preload"; as="style",</build/comment-css.css>; rel="preload"; as="style""
x-debug-token 
"1ac37a"

Cookies

Request Cookies

Key Value
PHPSESSID 
"vts6vkj800g6cuitjfcmnrirq3"

Response Cookies

No response cookies

Session 6

Session Metadata

Key Value
Created 
"Sat, 21 Jun 25 17:14:13 +0000"
Last used 
"Sat, 21 Jun 25 17:14:19 +0000"
Lifetime 
0

Session Attributes

Attribute Value
_csrf/https-comment 
"ztMkeF_pdrFHjiUFa2xHUiqyvq2MntTXG3FAuJuAWhw"
_csrf/https-cookie_accept 
"phMZinWxqjBpaRfYhWhkWQ8zLAUx8grrnFKSBWaYPio"

Session Usage

6 Usages
Stateless check enabled
Usage
Symfony\Component\Security\Csrf\TokenStorage\SessionTokenStorage:76
Show trace 
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Symfony\Component\Security\Csrf\TokenStorage\SessionTokenStorage:79
Show trace 
[
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Symfony\Component\Security\Csrf\TokenStorage\SessionTokenStorage:52
Show trace 
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Symfony\Component\Security\Csrf\TokenStorage\SessionTokenStorage:56
Show trace 
[
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Symfony\Component\Security\Core\Authentication\Token\Storage\UsageTrackingTokenStorage:41
Show trace 
[
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Symfony\Component\Security\Http\Firewall\ContextListener:171
Show trace 
[
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Flashes

Flashes

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Server Parameters

Server Parameters

Defined in .env

Key Value
APP_ENV 
"dev"
APP_SECRET 
"0a988e63f011514eaabfc650b599af4d"
CORS_ALLOW_ORIGIN 
"*"
DATABASE_URL 
"mysql://bbndb_rctuser:33F5W25z40or0f7@localhost:3306/rct_bbntimes"
GOOGLE_RECAPTCHA_SECRET_KEY 
"6LdV5fgpAAAAANxzTG8ZMfIjil1wu-1vrQvnUt-x"
GOOGLE_RECAPTCHA_SITE_KEY 
"6LdV5fgpAAAAAENKcn73MJAhQrbtQeqgyC4wDLMP"
MAILER_DSN 
"smtp://no-reply%40rct.dev.bbntimes.com:Bl6%26gLD48%26Of919@rct.dev.bbntimes.com:465"
MARIADB_PASSWORD 
"i!87pK&!85ezc8"

Defined as regular env variables

Key Value
APP_DEBUG 
"1"
BASE 
"/public"
CONTEXT_DOCUMENT_ROOT 
"/var/www/vhosts/dev.bbntimes.com/rct.dev.bbntimes.com"
CONTEXT_PREFIX 
""
DOCUMENT_ROOT 
"/var/www/vhosts/dev.bbntimes.com/rct.dev.bbntimes.com"
FCGI_ROLE 
"RESPONDER"
GATEWAY_INTERFACE 
"CGI/1.1"
HTTPS 
"on"
HTTP_ACCEPT 
"*/*"
HTTP_ACCEPT_ENCODING 
"gzip, br, zstd, deflate"
HTTP_CONNECTION 
"close"
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HTTP_HOST 
"rct.dev.bbntimes.com"
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"0"
PASSENGER_DOWNLOAD_NATIVE_SUPPORT_BINARY 
"0"
PATH 
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PHP_SELF 
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PP_CUSTOM_PHP_CGI_INDEX 
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PP_CUSTOM_PHP_INI 
"/var/www/vhosts/system/rct.dev.bbntimes.com/etc/php.ini"
QUERY_STRING 
""
REDIRECT_BASE 
"/public"
REDIRECT_HTTPS 
"on"
REDIRECT_PASSENGER_COMPILE_NATIVE_SUPPORT_BINARY 
"0"
REDIRECT_PASSENGER_DOWNLOAD_NATIVE_SUPPORT_BINARY 
"0"
REDIRECT_REDIRECT_HTTPS 
"on"
REDIRECT_REDIRECT_PASSENGER_COMPILE_NATIVE_SUPPORT_BINARY 
"0"
REDIRECT_REDIRECT_PASSENGER_DOWNLOAD_NATIVE_SUPPORT_BINARY 
"0"
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"rct.dev.bbntimes.com"
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"200"
REDIRECT_REDIRECT_UNIQUE_ID 
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REDIRECT_SSL_TLS_SNI 
"rct.dev.bbntimes.com"
REDIRECT_STATUS 
"200"
REDIRECT_UNIQUE_ID 
"aFboZ1DGsIl0EBapMk84bQAAANI"
REDIRECT_URL 
"/public/technology/the-tao-of-tau-intersection-of-tau-complex-numbers-code-libraries-and-javascript-in-analytics"
REMOTE_ADDR 
"216.73.216.240"
REMOTE_PORT 
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1750526056
REQUEST_TIME_FLOAT 
1750526056.8379
REQUEST_URI 
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SCRIPT_FILENAME 
"/var/www/vhosts/dev.bbntimes.com/rct.dev.bbntimes.com/public/index.php"
SCRIPT_NAME 
"/public/index.php"
SCRIPT_URI 
"https://rct.dev.bbntimes.com/technology/the-tao-of-tau-intersection-of-tau-complex-numbers-code-libraries-and-javascript-in-analytics"
SCRIPT_URL 
"/technology/the-tao-of-tau-intersection-of-tau-complex-numbers-code-libraries-and-javascript-in-analytics"
SERVER_ADDR 
"5.196.1.209"
SERVER_ADMIN 
"[no address given]"
SERVER_NAME 
"rct.dev.bbntimes.com"
SERVER_PORT 
"443"
SERVER_PROTOCOL 
"HTTP/1.1"
SERVER_SIGNATURE 
"<address>Apache Server at rct.dev.bbntimes.com Port 443</address>\n"
SERVER_SOFTWARE 
"Apache"
SSL_TLS_SNI 
"rct.dev.bbntimes.com"
SYMFONY_DOTENV_VARS 
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Sub Requests 6

CookieController :: renderAnalysis (token = 8eadd8)

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AppController :: renderHeader (token = c20e53)

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ArticleController :: trendingArticle (token = 4dd192)

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ArticleController :: relatedArticles (token = d51302)

Key Value
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"App\Controller\ArticleController::relatedArticles"
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"html"
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"en"
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current_article 
App\Entity\Article {#1094
  -id: 9311
  -title: "The Tao of Tau: Intersection of Tau, Complex Numbers, Code Libraries and Javascript in Analytics"
  -slug: "the-tao-of-tau-intersection-of-tau-complex-numbers-code-libraries-and-javascript-in-analytics"
  -introtext: "<p>One of the things that most young mathematicians (think eight years old or so) learn is that pie are square, a fact which is immediately refuted with the confused observation "No they're not!</p>\r\n"
  -content: """
    \r\n
    <p>Pies are round!" Later, of course, they discover algebra and geometry, where in fact it is proven demonstrably that, well, the circumference of a circle and the area of the same circle are related by two simple equations:</p>\r\n
    <p>C=2πr&nbsp;</p>\r\n
    <p>and</p>\r\n
    <p>A=πr^2</p>\r\n
    <p>The first law was at least known at the time of Euclid, though it would be a few centuries later that Archimedes actually calculated the ratio of the diameter (twice the radius) to the circumference to be roughly 3 1/7 (though in fact, he was able to get it right to within eight digits eventually, in Greek numeric notation). However, it would be early in the eighteen century AD that Welsh mathematician William Jones would use the Greek letter π (pi) for this particular ratio.</p>\r\n
    <p>The great mathematician Leonhard Euler would eventually pull together the use of π, the base of the natural logarithms (e, approximately 2.71828) and the square root of negative one (called i) into one of the more fundamental equations of mathematics:</p>\r\n
    <p>e^(πi) = -1</p>\r\n
    <p>For the next three hundred years, π would reign supreme in mathematics as perhaps the single most well known constant ever. However, for several years, Euler debated whether to use π to indicate the circumference of a circle divided by the diameter (a ratio of about 3.1415927...) or to indicate the circumference divided by the ratio (a ratio of 6.2831854...). His decision, ultimately, to go with the former may have complicated mathematics far more than he'd planned.</p>\r\n
    <p>In 2010, theoretical physicist Michael Hartl wrote a paper that would eventually become known as the Tau Manifesto, in which he proposed using the letter tau (τ) for the ratio of the circumference divided by radius, the aforementioned 6.28 value. His arguments were actually quite compelling:</p>\r\n
    <ul>\r\n
    <li>One&nbsp;<em>τ</em>&nbsp;is precisely one turn around a circle, or one revolution (360°).&nbsp;<em>τ</em>/2 is half a circle or 180°,&nbsp;<em>τ</em>/4 is one fourth of a circle, or 90° and so forth. Compare this with 2<em>π</em>,&nbsp;<em>π</em>, and&nbsp;<em>π</em>/2 respectively. In general, this becomes the same as C=&nbsp;<em>τ</em>r, as opposed to C= x<em>n</em>--</li>\r\n
    <li>Euler's equation (which laid the foundation for complex numbers), can be expresse<em>d</em>&nbsp;with&nbsp;τ&nbsp;as:</li>\r\n
    </ul>\r\n
    <p>e^(τi) = 1</p>\r\n
    <ul>\r\n
    <li>Similarly, the area of a circle takes on the same characteristics as other power laws in mathematics and physics:</li>\r\n
    </ul>\r\n
    <p>A = \dfrac{1}{2} \tau r^2; E= \dfrac{1}{2} m v^2;</p>\r\n
    <p>A=(1/2)τr^2 ; E = (1/2) mv^2</p>\r\n
    <p>Where it really shines, however, is in areas such as trigonometry and complex matrix algebra. For instance, if you had trigonometric functions built around τ, you can state that</p>\r\n
    <p>e^(iθ) = cos(θ) +&nbsp;<strong>i</strong>&nbsp;sin(θ), where 0 &lt; θ &lt; τ</p>\r\n
    <p>This equation makes it clear that a quarter of the way through a revolution, the equation has the value +<strong>i</strong>, at halfway it's -1, at 3/4 of the revolution, the equation is -<strong>i</strong>, and at one full revolution you're right back to where you started. It also highlights the close association between exponential functions and rotations.</p>\r\n
    <p><img src="/images/Pi_Chart.png" alt="Pi_Chart.png" width="601" height="568" /></p>\r\n
    <p>This becomes especially useful in gaming and 3D applications, in which the square root of -1 (the&nbsp;<strong><em>i</em></strong>&nbsp;part) is treated as an axis of rotation. In three dimensions, you end up with three such axes:&nbsp;<strong><em>i</em></strong>,&nbsp;<strong>j</strong><em>,&nbsp;</em>and&nbsp;<strong><em>k</em></strong>. These are known as&nbsp;<em>quaternions</em>. If each of these are used to describe a complex number (such as 0.5 + 0.866<strong><em>i</em></strong>) on the unit sphere (a great circle), then by performing a rotation on that circle (from 0 to 1 τ ), you can position that object precisely along that rotation. Camera operators (and pilots) understand these three dimensions as pitch, yaw and roll.</p>\r\n
    <p><img src="/images/Pitch_Axis.png" alt="Pitch_Axis.png" width="601" height="382" /></p>\r\n
    <p>These quaternions become especially important when dealing with rigging in three-dimensional work, involving vectors connecting end to end. These are also known as gimbals, again from the camera mounts that allow for orienting in any dimension. While in data science this can be done using matrix transformations, quaternion calculations are often faster and less time consuming to set up.</p>\r\n
    <p><img src="/images/Sin_O.png" alt="Sin_O.png" width="600" height="406" /></p>\r\n
    <p>The skeleton of a 3D figure is made up of vectors, with quaternions acting as gimbals at the joints to help move the figure around.</p>\r\n
    <p><img src="/images/A_Library_For_Tau_Based_Complex_Number_Operations.png" alt="A_Library_For_Tau_Based_Complex_Number_Operations.png" width="600" height="338" /></p>\r\n
    <h3>A Library For Tau Based Complex Number Operations</h3>\r\n
    <p>The value of Tau (and the relationship between Tau revolutions and complex numbers, inspired me to write a small Javascript library that both defined some Tau helper functions (the Tau class, made up exclusively of static) and an immutable TauComplex class that was built specifically with Tau revolutions in mind. These scripts are available at&nbsp;<a href="https://github.com/kurtcagle/tau" target="_blank" rel="nofollow noopener">https://github.com/kurtcagle/tau</a>.</p>\r\n
    <p><img src="/images/Class_Tau_1.png" alt="Class_Tau_1.png" width="600" height="582" /></p>\r\n
    <p><img src="/images/Clauss_Tau_2.png" alt="Clauss_Tau_2.png" width="600" height="618" /></p>\r\n
    <p><img src="/images/Clauss_Tau_3.png" alt="Clauss_Tau_3.png" width="600" height="598" /></p>\r\n
    <p><img src="/images/Clauss_Tau_3_-.png" alt="Clauss_Tau_3_-.png" width="601" height="201" /></p>\r\n
    <p>The complex class is reasonably complete for complex number manipulation, including handling addition and multiplication of complex numbers, creating complex conjugates, moduli, and equations for rotating, scaling and translating such numbers in the complex plane. A test script (TauTest.js) illustrates how these functions are invoked:</p>\r\n
    <p><img src="/images/console_log.png" alt="console_log.png" width="601" height="425" /></p>\r\n
    <p>with this test script generating the following output :</p>\r\n
    <p><img src="/images/C1-1.png" alt="C1-1.png" width="601" height="434" /></p>\r\n
    <p>One of the critical points about this library is that the complex numbers are meant to be treated as immutable, with any operation on a complex number generating a new complex number. For instance, if you wanted to add three complex numbers, you'd create an expression such as:</p>\r\n
    <p><img src="/images/const_c6.png" alt="const_c6.png" width="603" height="69" /></p>\r\n
    <p>Complex numbers factor heavily in both pure and applied mathematics, and can be proved to be the largest complete set of numbers - you cannot construct another space of numbers that cannot be decomposed into complex numbers. Additionally, the equations used to create approximations of surfaces, known as the Taylor Series, makes extensive use of complex numbers.</p>\r\n
    <p>There is one final point to be made in this particular article. There is a tendency to look at either Python or R when dealing with mathematics, but it's worth noting that it is perfectly possible to create mathematics libraries and classes in JavaScript as well. Indeed, Javascript contains its own equivalents of both Pandas (Danfo.js) and NumPy (nympy.js), two of the foundational classes, and Google's TensorFlow library has been ported over to Javascript as well (primarily around the node.js version). In some cases, Javascript is even faster than Python in the analytics space, especially given recent optimizations for handling binary data types.</p>
    """
  -user: Proxies\__CG__\App\Entity\User {#1092 …}
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    date: 2021-08-12 20:59:28.0 UTC (+00:00)
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  -metakey: "The Tao of Tau: Intersection of Tau, Complex Numbers, Code Libraries and Javascript in Analytics"
  -metadesc: "One of the things that most young mathematicians (think eight years old or so) learn is that pie are square, a fact which is immediately refuted with the confused observation "No they're not!"
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  -publishedAt: DateTime @1628801968 {#1147
    date: 2021-08-12 20:59:28.0 UTC (+00:00)
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AppController :: renderFooter (token = 29daf8)

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CookieController :: renderCookie (token = 4aac8e)

Key Value
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"App\Controller\CookieController::renderCookie"
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"html"
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"en"
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"ac353a"