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Chaos Theory

Snow Crystals, Fractals and Self Similarity

Introduction

The pieces of snow falling to the ground in winter have two very special scientific characteristics:

No two flakes are exactly alike
If you cut one, the remaining portions will have the same shape as what you cut from

These amazing qualities have been studied in the realm of Chaos Theory.

Uniqueness

According to a renowned scientist, finding two identical snowflakes is like ”shuffling a deck of cards and getting the exact same shuffle back. You could shuffle every second for the entire life of the universe, and you wouldn’t come close to getting two of the same”.¹ In order to explain this, we must look at the way that snow crystals are formed.

As snow is water, the basic blocks are in the majority of molecules two hydrogen atoms and one oxygen atom. While the atoms are all alike, about 1 in 5000 molecules² will be a so called ”heavy water” or ”deuterium oxide” compound, where the hydrogen atoms are the ²H isotope³ with two particles (one proton + one neutron) in the nucleus instead of the normal ¹H isotope.

These abberrations are placed at random points in the lattice. And since there are some 10¹⁸ atoms in a normal snowflake, about 10¹⁴ – 10¹⁵ deuterium molecules will be placed at some locations in the atomic structure.² The likelihood of two different configurations of such a high number being exactly the same is so low that it can be considered impossible!²

Self Similarity

The classical example of a system with self similarity is the Babushka or Matryoshka doll.⁴ When opening it, a doll with the exact shape and look comes out, only smaller, and this keeps repeating. The mathematical definition of this is a fractal, and this has been observed and used in many early cultural art forms.⁵

Snow is not perfectly self similar, but a mathematical fractal is like snow to a degree of 95-99 %.^6,7 It has been shown in a model of how snowflakes are formed that they indeed take a shape given by a fractal pattern.⁸

The Koch snowflake

Swedish researcher Helge von Koch used fractal theory to build figures of snowflakes as early as 1904. The process was to start with a triangle and then continuously split each side and add another triangle at the half point. This yielded the initial sequence of shapes shown in Figure 1.

Figure 1. Four first Koch snowflakes [9].

References

[1] Prof. Kenneth Libbrecht, Californa Institute of Technology, quote from PBS Newshour.

[2] http://www.its.caltech.edu/~atomic/snowcrystals/alike/alike.htm - Snowflakes - No Two Alike? Caltech.

[3] http://pubchem.ncbi.nlm.nih.gov/compound/deuterium_oxide - Open Chemistry Database.

[4] https://en.wikipedia.org/wiki/Matryoshka_doll - Wikipedia.

[5] http://polynomial.me.uk/2009/09/20/fractals-of-brain-fractals-of-mind/ - Self Similarity ~ Fractals, Fractals Everywhere, Karl Richard 2009.

[6] Fractals – A Brief Explanation. Junior University, available through http://juni.osfc.ac.uk/Aim_Higher/Maths/

[7] Prof. Craig A. Tovey, Georgia Institue of Technology, answered on http://www.researchgate.net/post/Are_snowflakes_based_on_fractal_geometry .

[8] Nittmann, J. and Stanley, H.E.: Non-deterministic approach to anisotropic growth patterns with continuously tunable morphology: the fractal properties of some real snowflakes, Journal of Physics A, vol. 20, Number 17, also available at http://iopscience.iop.org/0305-4470/20/17/010

[9]. http://mathworld.wolfram.com/KochSnowflake.html - Wolfram MathWorld.

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eloquent.retrieved: App\Models\Subject (9μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (9μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (9μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (9μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (9μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (9μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (9μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (9μs)
eloquent.retrieved: App\Models\Subject (8μs)
eloquent.retrieved: App\Models\Subject (9μs)
eloquent.retrieved: App\Models\Subject (8μs)
Illuminate\Database\Events\StatementPrepared (1.12ms)
Illuminate\Database\Events\QueryExecuted (953μs)
eloquent.retrieved: App\Models\SubjectCat (101μs)
eloquent.retrieved: App\Models\SubjectCat (18μs)
eloquent.retrieved: App\Models\SubjectCat (13μs)
eloquent.retrieved: App\Models\SubjectCat (10μs)
eloquent.retrieved: App\Models\SubjectCat (11μs)
eloquent.retrieved: App\Models\SubjectCat (9μs)
Illuminate\Cache\Events\KeyWritten (18.82ms)
Illuminate\Cache\Events\KeyWritten (30.15ms)
Illuminate\Cache\Events\CacheHit (18.09ms)
Illuminate\Cache\Events\CacheHit (1.39ms)
Illuminate\Cache\Events\CacheHit (205μs)
Illuminate\Cache\Events\CacheHit (193μs)
Illuminate\Cache\Events\CacheHit (157μs)
Illuminate\Cache\Events\CacheHit (207μs)
Illuminate\Cache\Events\CacheHit (158μs)
Illuminate\Cache\Events\CacheHit (175μs)
Illuminate\Cache\Events\CacheHit (151μs)
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Illuminate\Cache\Events\CacheHit (164μs)
Illuminate\Cache\Events\CacheHit (180μs)
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Illuminate\Cache\Events\CacheHit (182μs)
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Illuminate\Cache\Events\CacheHit (240μs)
Illuminate\Cache\Events\CacheHit (2.11ms)
Illuminate\Cache\Events\CacheHit (343μs)
Illuminate\Cache\Events\CacheHit (627μs)
Illuminate\Cache\Events\CacheHit (8.32ms)
Illuminate\Cache\Events\CacheHit (416μs)
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Illuminate\Cache\Events\CacheHit (210μs)
Illuminate\Cache\Events\CacheHit (262μs)
Illuminate\Cache\Events\CacheHit (835μs)
Illuminate\Cache\Events\CacheHit (295μs)
Illuminate\Cache\Events\CacheHit (583μs)
Illuminate\Cache\Events\CacheHit (710μs)
Illuminate\Cache\Events\CacheHit (8.55ms)
Illuminate\Cache\Events\CacheHit (408μs)
Illuminate\Cache\Events\CacheHit (259μs)
Illuminate\Cache\Events\CacheHit (176μs)
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Illuminate\Cache\Events\CacheHit (173μs)
Illuminate\Cache\Events\CacheHit (8.44ms)
Illuminate\Cache\Events\CacheHit (401μs)
Illuminate\Cache\Events\CacheHit (282μs)
Illuminate\Cache\Events\CacheHit (181μs)
Illuminate\Cache\Events\CacheHit (462μs)
Illuminate\Cache\Events\CacheHit (274μs)
Illuminate\Cache\Events\CacheHit (217μs)
Illuminate\Cache\Events\CacheHit (172μs)
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Illuminate\Cache\Events\CacheHit (203μs)
Illuminate\Cache\Events\CacheHit (154μs)
Illuminate\Cache\Events\CacheHit (245μs)
Illuminate\Cache\Events\CacheHit (277μs)
Illuminate\Cache\Events\CacheHit (203μs)
Illuminate\Cache\Events\CacheHit (163μs)
Illuminate\Cache\Events\CacheHit (20.37ms)
Illuminate\Cache\Events\CacheHit (340μs)
Illuminate\Cache\Events\CacheHit (218μs)
Illuminate\Cache\Events\CacheHit (167μs)
Illuminate\Cache\Events\CacheHit (197μs)
Illuminate\Cache\Events\CacheHit (157μs)
Illuminate\Cache\Events\CacheHit (195μs)
Illuminate\Cache\Events\CacheHit (250μs)
Illuminate\Cache\Events\CacheHit (254μs)
Illuminate\Cache\Events\CacheHit (171μs)
Illuminate\Cache\Events\CacheHit (269μs)
Illuminate\Cache\Events\CacheHit (166μs)
Illuminate\Cache\Events\CacheHit (301μs)
Illuminate\Cache\Events\CacheHit (180μs)
Illuminate\Cache\Events\CacheHit (196μs)
Illuminate\Cache\Events\CacheHit (163μs)
Illuminate\Cache\Events\CacheHit (266μs)
Illuminate\Cache\Events\CacheHit (364μs)
Illuminate\Cache\Events\CacheHit (224μs)
Illuminate\Cache\Events\CacheHit (159μs)
Illuminate\Cache\Events\CacheHit (176μs)
Illuminate\Cache\Events\CacheHit (147μs)
Illuminate\Cache\Events\CacheHit (408μs)
Illuminate\Cache\Events\CacheHit (286μs)
Illuminate\Cache\Events\CacheHit (198μs)
Illuminate\Cache\Events\CacheHit (269μs)
Illuminate\Cache\Events\CacheHit (364μs)
Illuminate\Cache\Events\CacheHit (203μs)
Illuminate\Cache\Events\CacheHit (398μs)
Illuminate\Cache\Events\CacheHit (202μs)
Illuminate\Cache\Events\CacheHit (196μs)
Illuminate\Cache\Events\CacheHit (161μs)
Illuminate\Cache\Events\CacheHit (202μs)
Illuminate\Cache\Events\CacheHit (175μs)
Illuminate\Cache\Events\CacheHit (200μs)
Illuminate\Cache\Events\CacheHit (187μs)
Illuminate\Cache\Events\CacheHit (185μs)
Illuminate\Cache\Events\CacheHit (157μs)
Illuminate\Cache\Events\CacheHit (183μs)
Illuminate\Cache\Events\CacheHit (158μs)
Illuminate\Cache\Events\CacheHit (185μs)
Illuminate\Cache\Events\CacheHit (154μs)
Illuminate\Cache\Events\CacheHit (185μs)
Illuminate\Cache\Events\CacheHit (155μs)
Illuminate\Cache\Events\CacheHit (181μs)
Illuminate\Cache\Events\CacheHit (156μs)
Illuminate\Cache\Events\CacheHit (184μs)
Illuminate\Cache\Events\CacheHit (247μs)
Illuminate\Cache\Events\CacheHit (197μs)
Illuminate\Cache\Events\CacheHit (193μs)
Illuminate\Cache\Events\CacheHit (184μs)
Illuminate\Cache\Events\CacheHit (156μs)
creating: site.blocks.subject-homework (21.2ms)
composing: site.blocks.subject-homework (31μs)
eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (625μs)
eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (322μs)
Illuminate\Database\Events\StatementPrepared (9.82ms)
Illuminate\Database\Events\QueryExecuted (1.47ms)
creating: components.homework-blocks (189μs)
composing: components.homework-blocks (121μs)
creating: site.blocks.reviews (2.74ms)
composing: site.blocks.reviews (23μs)
creating: site.blocks.submit-homework (161μs)
composing: site.blocks.submit-homework (19μs)
creating: site.blocks.faq (129μs)
composing: site.blocks.faq (8.11ms)
eloquent.booting: App\Models\Twill\TwillFaq (431μs)
eloquent.booted: App\Models\Twill\TwillFaq (305μs)
eloquent.booting: App\Models\Twill\Slugs\TwillFaqSlug (339μs)
eloquent.booted: App\Models\Twill\Slugs\TwillFaqSlug (77μs)
Illuminate\Database\Events\StatementPrepared (7.26ms)
Illuminate\Database\Events\QueryExecuted (1.09ms)
eloquent.retrieved: App\Models\Twill\TwillFaq (174μs)
Illuminate\Database\Events\StatementPrepared (729μs)
Illuminate\Database\Events\QueryExecuted (1.18ms)
eloquent.retrieved: App\Models\Twill\Slugs\TwillFaqSlug (114μs)
eloquent.retrieved: App\Models\Twill\Slugs\TwillFaqSlug (22μs)
creating: components.faq-item (371μs)
composing: components.faq-item (118μs)
Illuminate\Database\Events\StatementPrepared (1.35ms)
Illuminate\Database\Events\QueryExecuted (1.04ms)
eloquent.retrieved: App\Models\Twill\TwillFaq (117μs)
Illuminate\Database\Events\StatementPrepared (5.33ms)
Illuminate\Database\Events\QueryExecuted (4.61ms)
eloquent.retrieved: App\Models\Twill\Slugs\TwillFaqSlug (124μs)
eloquent.retrieved: App\Models\Twill\Slugs\TwillFaqSlug (24μs)
creating: components.faq-item (898μs)
composing: components.faq-item (114μs)
Illuminate\Database\Events\StatementPrepared (1.31ms)
Illuminate\Database\Events\QueryExecuted (1.43ms)
eloquent.retrieved: App\Models\Twill\TwillFaq (129μs)
Illuminate\Database\Events\StatementPrepared (3.31ms)
Illuminate\Database\Events\QueryExecuted (4.67ms)
eloquent.retrieved: App\Models\Twill\Slugs\TwillFaqSlug (189μs)
creating: components.faq-item (106μs)
composing: components.faq-item (55μs)
Illuminate\Database\Events\StatementPrepared (1.24ms)
Illuminate\Database\Events\QueryExecuted (912μs)
eloquent.retrieved: App\Models\Twill\TwillFaq (109μs)
Illuminate\Database\Events\StatementPrepared (613μs)
Illuminate\Database\Events\QueryExecuted (1.01ms)
eloquent.retrieved: App\Models\Twill\Slugs\TwillFaqSlug (70μs)
creating: components.faq-item (82μs)
composing: components.faq-item (52μs)
creating: site.blocks.text-with-image (287μs)
composing: site.blocks.text-with-image (21μs)
creating: site.blocks.work-with-tutors (275μs)
composing: site.blocks.work-with-tutors (22μs)
creating: site.layouts.app (233μs)
composing: site.layouts.app (19μs)
creating: components.canonical (365μs)
composing: components.canonical (1.26ms)
creating: components.open-graph (238μs)
composing: components.open-graph (83μs)
creating: site.headers.header (427μs)
composing: site.headers.header (85μs)
Illuminate\Cache\Events\CacheHit (3.12ms)
creating: components.footer (84μs)
composing: components.footer (93μs)
Illuminate\Cache\Events\CacheHit (10.08ms)
Illuminate\Cache\Events\CacheHit (678μs)
Illuminate\Cache\Events\CacheHit (689μs)
Illuminate\Cache\Events\CacheHit (248μs)
Illuminate\Cache\Events\CacheHit (376μs)
Illuminate\Cache\Events\CacheHit (515μs)
Illuminate\Cache\Events\CacheHit (256μs)
Illuminate\Cache\Events\CacheHit (468μs)
Illuminate\Cache\Events\CacheHit (493μs)
Illuminate\Cache\Events\CacheHit (236μs)
Illuminate\Cache\Events\CacheHit (202μs)
Illuminate\Cache\Events\CacheHit (366μs)
Illuminate\Cache\Events\CacheHit (202μs)
Illuminate\Cache\Events\CacheHit (390μs)
creating: components.forms.contact-us (457μs)
composing: components.forms.contact-us (208μs)
creating: components.forms.get-started (169μs)
composing: components.forms.get-started (8.24ms)
creating: components.forms.free-tool-download (494μs)
composing: components.forms.free-tool-download (184μs)
creating: components.forms.claim-free-worksheet (166μs)
composing: components.forms.claim-free-worksheet (77μs)
creating: components.forms.tutor-subscription-waitlist (103μs)
composing: components.forms.tutor-subscription-waitlist (69μs)
creating: components.forms.tutor-subscription-join (99μs)
composing: components.forms.tutor-subscription-join (66μs)
creating: components.forms.tutor-support (102μs)
composing: components.forms.tutor-support (65μs)

381 x Illuminate\Cache\Events\CacheHit (25.43%)	320ms
21 x Illuminate\Database\Events\StatementPrepared (7.33%)	92.31ms
21 x Illuminate\Database\Events\QueryExecuted (4.89%)	61.56ms
4 x Illuminate\Cache\Events\KeyWritten (4.76%)	59.94ms
1 x creating: site.blocks.subject-homework (1.68%)	21.20ms
1 x Illuminate\Foundation\Events\LocaleUpdated (0.73%)	9.15ms
1 x composing: components.forms.get-started (0.65%)	8.24ms
1 x composing: site.blocks.faq (0.64%)	8.11ms
1 x Illuminate\Routing\Events\RouteMatched (0.28%)	3.55ms
2 x Illuminate\Database\Events\ConnectionEstablished (0.22%)	2.77ms
1 x creating: site.blocks.reviews (0.22%)	2.74ms
3 x creating: components.tutors-card (0.22%)	2.71ms
1 x Illuminate\Routing\Events\Routing (0.19%)	2.40ms
4 x creating: components.faq-item (0.12%)	1.46ms
96 x eloquent.retrieved: App\Models\Subject (0.11%)	1.33ms
1 x composing: components.canonical (0.1%)	1.26ms
4 x Illuminate\Cache\Events\CacheMissed (0.05%)	682μs
1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)	625μs
6 x eloquent.retrieved: App\Models\Twill\Slugs\TwillFaqSlug (0.04%)	543μs
1 x eloquent.booting: App\Models\User\TutorAttributes\TutorDegree (0.04%)	536μs
4 x eloquent.retrieved: App\Models\Twill\TwillFaq (0.04%)	529μs
1 x creating: components.forms.free-tool-download (0.04%)	494μs
1 x creating: site.blocks.how-it-works (0.04%)	482μs
1 x creating: components.forms.contact-us (0.04%)	457μs
1 x eloquent.booting: App\Models\Twill\TwillFaq (0.03%)	431μs
1 x creating: site.headers.header (0.03%)	427μs
9 x eloquent.retrieved: App\Models\SubjectCat (0.03%)	404μs
1 x creating: components.canonical (0.03%)	365μs
1 x eloquent.booting: App\Models\Subject (0.03%)	349μs
1 x creating: components.tutors-cards (0.03%)	346μs
1 x eloquent.booting: App\Models\Twill\Slugs\TwillFaqSlug (0.03%)	339μs
4 x composing: components.faq-item (0.03%)	339μs
1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)	322μs
1 x eloquent.booting: App\Models\User\Tutor (0.03%)	320μs
1 x eloquent.booted: App\Models\Twill\TwillFaq (0.02%)	305μs
1 x creating: components.breadcrumbs (0.02%)	294μs
1 x creating: site.blocks.text-with-image (0.02%)	287μs
1 x creating: site.blocks.work-with-tutors (0.02%)	275μs
3 x composing: components.tutors-card (0.02%)	262μs
1 x creating: components.open-graph (0.02%)	238μs
1 x creating: site.layouts.app (0.02%)	233μs
1 x composing: subjects.show (0.02%)	225μs
1 x composing: components.forms.contact-us (0.02%)	208μs
1 x eloquent.booted: App\Models\User\Tutor (0.02%)	204μs
1 x creating: components.homework-blocks (0.02%)	189μs
1 x composing: components.forms.free-tool-download (0.01%)	184μs
1 x eloquent.booting: App\Models\Meta (0.01%)	181μs
3 x eloquent.retrieved: App\Models\User\Tutor (0.01%)	179μs
1 x creating: subjects.show (0.01%)	178μs
1 x creating: components.forms.get-started (0.01%)	169μs
1 x creating: components.forms.claim-free-worksheet (0.01%)	166μs
1 x creating: site.blocks.submit-homework (0.01%)	161μs
1 x creating: site.blocks.subject-tutors (0.01%)	154μs
1 x composing: components.breadcrumbs (0.01%)	146μs
1 x creating: site.blocks.faq (0.01%)	129μs
1 x composing: components.tutors-cards (0.01%)	124μs
1 x composing: components.homework-blocks (0.01%)	121μs
1 x eloquent.booting: App\Models\SubjectCat (0.01%)	106μs
1 x creating: components.forms.tutor-subscription-waitlist (0.01%)	103μs
1 x creating: components.forms.tutor-support (0.01%)	102μs
1 x creating: components.forms.tutor-subscription-join (0.01%)	99μs
1 x eloquent.retrieved: App\Models\Meta (0.01%)	96μs
1 x composing: components.footer (0.01%)	93μs
1 x composing: site.headers.header (0.01%)	85μs
1 x creating: components.footer (0.01%)	84μs
1 x composing: components.open-graph (0.01%)	83μs
1 x eloquent.booted: App\Models\Twill\Slugs\TwillFaqSlug (0.01%)	77μs
1 x composing: components.forms.claim-free-worksheet (0.01%)	77μs
1 x eloquent.booted: App\Models\Subject (0.01%)	69μs
1 x composing: components.forms.tutor-subscription-waitlist (0.01%)	69μs
1 x composing: components.forms.tutor-subscription-join (0.01%)	66μs
1 x eloquent.booted: App\Models\User\TutorAttributes\TutorDegree (0.01%)	66μs
1 x composing: components.forms.tutor-support (0.01%)	65μs
1 x eloquent.booted: App\Models\SubjectCat (0%)	61μs
1 x eloquent.booted: App\Models\Meta (0%)	60μs
1 x composing: site.blocks.subject-homework (0%)	31μs
1 x composing: site.blocks.how-it-works (0%)	24μs
1 x composing: site.blocks.reviews (0%)	23μs
1 x composing: site.blocks.work-with-tutors (0%)	22μs
1 x composing: site.blocks.text-with-image (0%)	21μs
1 x composing: site.blocks.subject-tutors (0%)	20μs
1 x composing: site.blocks.submit-homework (0%)	19μs
1 x composing: site.layouts.app (0%)	19μs

31 templates were rendered

1x subjects.showshow.blade.phpblade
1x components.breadcrumbsbreadcrumbs.blade.phpblade
1x site.blocks.how-it-workshow-it-works.blade.phpblade
1x site.blocks.subject-tutorssubject-tutors.blade.phpblade
1x components.tutors-cardstutors-cards.blade.phpblade
3x components.tutors-cardtutors-card.blade.phpblade
1x site.blocks.subject-homeworksubject-homework.blade.phpblade
1x components.homework-blockshomework-blocks.blade.phpblade
1x site.blocks.reviewsreviews.blade.phpblade
1x site.blocks.submit-homeworksubmit-homework.blade.phpblade
1x site.blocks.faqfaq.blade.phpblade
4x components.faq-itemfaq-item.blade.phpblade
1x site.blocks.text-with-imagetext-with-image.blade.phpblade
1x site.blocks.work-with-tutorswork-with-tutors.blade.phpblade
1x site.layouts.appapp.blade.phpblade
1x components.canonicalcanonical.blade.phpblade
1x components.open-graphopen-graph.blade.phpblade
1x site.headers.headerheader.blade.phpblade
1x components.footerfooter.blade.phpblade
1x components.forms.contact-uscontact-us.blade.phpblade
1x components.forms.get-startedget-started.blade.phpblade
1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
1x components.forms.tutor-supporttutor-support.blade.phpblade

uri: GET subjects/{categoryUri}/{subjectUri?}
middleware: web, utm.parameters
controller: App\Http\Controllers\SubjectController@show
namespace
where
as: subjects.show
file: app/Http/Controllers/SubjectController.php:43-96

21 statements were executed127ms

Connection Establishedtwenty4_siteSubjectRepository.php#123

Backtrace

13. app/Repositories/SubjectRepository.php:123
14. app/Http/Controllers/SubjectController.php:54
15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260

select * from `subjects` where `urlFriendlyTitle` = 'Chaos-Theory' limit 1

19.93mstwenty4_siteSubjectRepository.php#123

Bindings	0: Chaos-Theory
Backtrace	15. app/Repositories/SubjectRepository.php:123 16. app/Http/Controllers/SubjectController.php:54 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260

select * from `subject_cats` where `subject_cats`.`id` in (34)

1.82mstwenty4_siteSubjectRepository.php#123

Backtrace

20. app/Repositories/SubjectRepository.php:123
21. app/Http/Controllers/SubjectController.php:54
22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260

select * from `meta` where `meta`.`id` = 3 limit 1

1.38mstwenty4_siteMetaService.php#118

Bindings	0: 3
Backtrace	19. app/Services/MetaService.php:118 20. app/Services/MetaService.php:108 21. app/Services/MetaService.php:81 22. app/Http/Controllers/SubjectController.php:89 23. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54

Connection Establishedtwenty4_siteTutorRepository.php#40

Backtrace

13. app/Repositories/TutorRepository.php:40
14. app/View/Components/TutorsCards.php:28
15. app/View/Components/TutorsCards.php:22
17. vendor/laravel/framework/src/Illuminate/Container/Container.php:929
18. vendor/laravel/framework/src/Illuminate/Container/Container.php:770

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21.18mstwenty4_siteTutorsCards.php#35

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select * from `users` where (`subjectsQualified` like '%|417|%' or `DSF_subjectsQualified` like '%|417|%' or `allsubjects` = 1) and `disable_profile` = 0 and `picture` != '' and `solutionrequestQualified` = 1 and `is_suspended` = 0 and `is_deactivated` = 0 and `status` != '' and `status` = 'tutor' order by `completed_requests` desc limit 6 offset 0

11.81mstwenty4_siteTutorsCards.php#35

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900μstwenty4_siteTutor.php#361

Bindings	0: 76351
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6.88mstwenty4_siteTutor.php#361

Bindings	0: 109553
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5.02mstwenty4_siteTutor.php#214

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1.19mstwenty4_siteTutor.php#214

Backtrace

19. app/Models/User/Tutor.php:214
20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
23. app/Models/User/Tutor.php:216
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select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 104601 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

10.21mstwenty4_siteTutor.php#361

Bindings	0: 104601
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select `subjects`.*, `subject_cats`.`title` as `category_title`, `subject_cats`.`id` as `category_id` from `subjects` inner join `subject_cats` on `subject_cats`.`id` = `subjects`.`catID` where `subjects`.`id` in (515, 459, 516, 517, 227, 30, 293, 294, 208, 250, 26, 211, 212, 210, 240, 241, 119, 34, 2, 160, 1, 6, 127, 123, 229, 70, 10, 122, 257, 220, 126, 124, 65, 7, 3, 349, 9, 66, 125, 68, 47, 73, 348, 41, 8, 347, 67, 48, 121, 4, 223, 417, 401, 325, 479, 14, 379, 226, 327, 501, 116, 147, 89, 446, 88, 269)

7.3mstwenty4_siteTutor.php#214

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14. app/Models/User/Tutor.php:214
15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
18. app/Models/User/Tutor.php:216
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select * from `subject_cats` where `subject_cats`.`id` in (1, 5, 10, 31, 34, 42)

1.13mstwenty4_siteTutor.php#214

Backtrace

19. app/Models/User/Tutor.php:214
20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
23. app/Models/User/Tutor.php:216
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1.21mstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:7
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (2) and `twill_faq_slugs`.`deleted_at` is null

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Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:24
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'how-much-will-it-cost' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

1.03mstwenty4_siteFaqItem.php#15

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Backtrace	15. app/View/Components/FaqItem.php:15 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.faq:41 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (3) and `twill_faq_slugs`.`deleted_at` is null

6.98mstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:41
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'what-kind-of-payments-do-you-accept' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

940μstwenty4_siteFaqItem.php#15

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1.02mstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:58
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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