mandelbrot/tools/deep-zoom/LOCATIONS.md

Deep-zoom locations

Beautiful deep Mandelbrot locations, all computed with find_deep.py and checked with a --headless render. Every one is an exact minibrot nucleus (Newton's method at 2·depth + 60 digits), not a coordinate copied from somewhere, so the digits are right all the way down.

How to use them

Each location has two views on the same centre:

• Minibrot: half-height ≈ 3× the minibrot's size. A tiny copy of the whole set sits in concentric colour bands, and there are more bands the deeper it is.
• Embedded Julia set: half-height ≈ size^0.75, about ¾ of the way down in log-zoom. This is where each location looks unique. The minibrots all look alike.

cargo run --release -- --view=RE,IM,HALF_HEIGHT        # interactive
cargo run --release -- --headless --view=RE,IM,HALF_HEIGHT --export-path shot.png
# zoom video from the full set down to a minibrot, piped to ffmpeg (see Makefile):
cargo run --release -- --headless --view=-0.75,0,1.5 --to-view=RE,IM,HALF_HEIGHT \
    --fps 60 --duration 60 --export-path - | make hevc-nvenc

Pass negative coordinates as --view=-0.77... (with the =): clap would read --view -0.77... as a flag. Iterations scale with depth automatically (about 90k at 10⁻¹⁰⁰ and 270k at 10⁻³⁰⁰), which is enough for these periods.

Infinite spirals (Misiurewicz points)

These are the anchor points. Zoom straight into one and the same spiral repeats forever, at any depth. Every minibrot below sits just beside one of them.

Region	Point	Preperiod, period	Looks like
Seahorse valley	-0.7756837680090537974694835039347410457282465046158 0.13646736829469012473327440961784876519777211159247	24, 2	Two-armed spirals of seahorses, repeating at every depth.
Elephant valley	0.29207386191445391799971751769220573408040189143315 0.015770917211390118082469028230338337682600488632339	32, 3	Three-armed spirals of elephant trunks.
Top antenna	-0.10110375797210418675706890936004723524958119917843 0.95629400795235409290719434118824484558046591980423	38, 4	Four-armed branch points on the upper antenna, with thin dendrites instead of spirals.

Seahorse valley

Seahorse valley, minibrot at 3.7e-41 (period 1034)

The embedded Julia set is a pale double spiral: two S-shaped arms of tiny seahorses wind into a bright centre, with a third lace band sweeping around them. The best one for a short video, and it renders in seconds.

# embedded Julia set
--view=-0.77568376800905379745347652613832924487504096622022,0.13646736829469012473375311880735014411233827361594,1e-30
# minibrot
--view=-0.77568376800905379745347652613832924487504096622022,0.13646736829469012473375311880735014411233827361594,1.1e-40

Seahorse valley, minibrot at 1.14e-101 (period 2868)

At the Julia depth there's a dark blue field with a glowing core and lace threads crossing it.

# embedded Julia set
--view=-0.77568376800905379746948350393474104572824650461579342765769586746577714101553375163935420167637213972439444483,0.13646736829469012473327440961784876519777211159246218448277041663416833530840062457397927487282583495378053495,1e-76
# minibrot
--view=-0.77568376800905379746948350393474104572824650461579342765769586746577714101553375163935420167637213972439444483,0.13646736829469012473327440961784876519777211159246218448277041663416833530840062457397927487282583495378053495,3.4e-101

Seahorse valley, minibrot at 1.37e-200 (period 5895)

Two filaments cross in an X over a warm brown background, with a small dark spiral where they meet.

# embedded Julia set
--view=-0.77568376800905379746948350393474104572824650461580039479495200140066601722281958989488315936878225494657853849784368247991666558116221905228489656259693587754973261632614186930985854124920967267440737850057557,0.13646736829469012473327440961784876519777211159246769640780019540083566597481550087544861124417865661202932094564677695883049019713653069907950852152823160218141353878979871215911794286224832089496659451498185,1e-150
# minibrot
--view=-0.77568376800905379746948350393474104572824650461580039479495200140066601722281958989488315936878225494657853849784368247991666558116221905228489656259693587754973261632614186930985854124920967267440737850057557,0.13646736829469012473327440961784876519777211159246769640780019540083566597481550087544861124417865661202932094564677695883049019713653069907950852152823160218141353878979871215911794286224832089496659451498185,4.1e-200

Seahorse valley, minibrot at 1.83e-300 (period 8922)

Faint cream threads, and a spiral wound so tight it reads as a point. Period 8922, so the minibrot needs a long render.

# embedded Julia set
--view=-0.775683768009053797469483503934741045728246504615800394794952001400666017222819589894883159368782255028252296689050972508942991790804637379375064948276511044167168394677127466362472192356953918438738697548775738756167744421444715134058363508346750264411321139485834882343906172015818621107808449401958194265462,0.136467368294690124733274409617848765197772111592467696407800195400835665974815500875448611244178656625706823059759836009902469392551108149827753781145338163746214967880988693412636771323356061132825037923307975359146666033691080243216637006179374407866237575096209504710178751614036006531433101633993532313930,1e-225
# minibrot
--view=-0.775683768009053797469483503934741045728246504615800394794952001400666017222819589894883159368782255028252296689050972508942991790804637379375064948276511044167168394677127466362472192356953918438738697548775738756167744421444715134058363508346750264411321139485834882343906172015818621107808449401958194265462,0.136467368294690124733274409617848765197772111592467696407800195400835665974815500875448611244178656625706823059759836009902469392551108149827753781145338163746214967880988693412636771323356061132825037923307975359146666033691080243216637006179374407866237575096209504710178751614036006531433101633993532313930,5.5e-300

Elephant valley

Elephant valley, minibrot at 1.29e-100 (period 782)

A cream-coloured double S-spiral, like the seahorse one but with fatter, beaded arms.

# embedded Julia set
--view=0.2920738619144539179997175176922057340804018914331633820622644536373359023331882946705674169275530423410817910,0.01577091721139011808246902823033833768260048863232968822435568049949556139801802963232184866659058960379114747,1e-75
# minibrot
--view=0.2920738619144539179997175176922057340804018914331633820622644536373359023331882946705674169275530423410817910,0.01577091721139011808246902823033833768260048863232968822435568049949556139801802963232184866659058960379114747,3.9e-100

Elephant valley, minibrot at 1.43e-199 (period 1585)

A single huge spiral arm made of blue beads curling across the whole frame.

# embedded Julia set
--view=0.2920738619144539179997175176922057340804018914331538397786950546981063169810374784665382516408545756138262706218874924432355556243165250541700374639889797148498919340003273815669351046658761147565052417000965,0.01577091721139011808246902823033833768260048863233880759961760295375606839216565332458201188913912642954367433579737643167232896069795830226759356080847862646789481028238105648099709502522309228500630759855757,1e-149
# minibrot
--view=0.2920738619144539179997175176922057340804018914331538397786950546981063169810374784665382516408545756138262706218874924432355556243165250541700374639889797148498919340003273815669351046658761147565052417000965,0.01577091721139011808246902823033833768260048863233880759961760295375606839216565332458201188913912642954367433579737643167232896069795830226759356080847862646789481028238105648099709502522309228500630759855757,4.3e-199

Elephant valley, minibrot at 3.82e-300 (period 2395)

A spiral galaxy: several beaded arms around a pinwheel centre. One of the prettiest frames here.

# embedded Julia set
--view=0.292073861914453917999717517692205734080401891433153839778695054698106316981037478466538251640854575589843066619974864800862332115945960921998657430606126553202998390050225413256445451049563878137878839982267258218894836912449161037745627504826537881261626243871678392881757578278459983842286413975991353319739,0.0157709172113901180824690282303383376826004886323388075996176029537560683921656533245820118891391263013273917522825544701901299061542128984468742890308365571942081484001405437827548345359998503228665872823816067983725089244499770271352539008763406229496083105636482420467114597267096082878888045800333807911146,1e-225
# minibrot
--view=0.292073861914453917999717517692205734080401891433153839778695054698106316981037478466538251640854575589843066619974864800862332115945960921998657430606126553202998390050225413256445451049563878137878839982267258218894836912449161037745627504826537881261626243871678392881757578278459983842286413975991353319739,0.0157709172113901180824690282303383376826004886323388075996176029537560683921656533245820118891391263013273917522825544701901299061542128984468742890308365571942081484001405437827548345359998503228665872823816067983725089244499770271352539008763406229496083105636482420467114597267096082878888045800333807911146,1.1e-299

Top antenna

Top antenna, minibrot at 6.11e-101 (period 245)

A dark branching dendrite like a frost cross, with four-way junctions.

# embedded Julia set
--view=-0.10110375797210418675706890936004723524958119917842627217950669362738928440088634862532432662296073501093701563,0.95629400795235409290719434118824484558046591980423293884325556254155313535666457147658005358708674409029391915,1e-75
# minibrot
--view=-0.10110375797210418675706890936004723524958119917842627217950669362738928440088634862532432662296073501093701563,0.95629400795235409290719434118824484558046591980423293884325556254155313535666457147658005358708674409029391915,1.8e-100

Top antenna, minibrot at 5.28e-200 (period 473)

A chain of six-armed snowflakes linked by thin filaments.

# embedded Julia set
--view=-0.10110375797210418675706890936004723524958119917842839398761868801597328463328677019373309617158827470368429189213012997868613724029746440432852247903869013157556141380437202406412221853553594880182186959159214,0.95629400795235409290719434118824484558046591980423399388630140415938010604675760901574969597486831722998038561988250849596064882005832681021199867736493316924923123916964900533995659228499205141785254396348248,1e-149
# minibrot
--view=-0.10110375797210418675706890936004723524958119917842839398761868801597328463328677019373309617158827470368429189213012997868613724029746440432852247903869013157556141380437202406412221853553594880182186959159214,0.95629400795235409290719434118824484558046591980423399388630140415938010604675760901574969597486831722998038561988250849596064882005832681021199867736493316924923123916964900533995659228499205141785254396348248,1.6e-199

Top antenna, minibrot at 8.34e-301 (period 705)

A single twelve-armed star, very symmetric. The antenna minibrots have short periods (245-705), so these are the fastest deep renders.

# embedded Julia set
--view=-0.1011037579721041867570689093600472352495811991784283939876186880159732846332867701937330961715882746718394428627914481009391257007785408536971203165750666758077852384490279260184029373211737264310068845597273991782305778295187471341711226626762421910644338164121042775094463557363347706110614218633830535086184,0.9562940079523540929071943411882448455804659198042339938863014041593801060467576090157496959748683171680339080646917869452781885502668216669916185990922330667568398963038228431738928191227884090004548615469303071899797403388188723954061899156655653387543351937028936674153437083087070219108726088551230849901498,1e-225
# minibrot
--view=-0.1011037579721041867570689093600472352495811991784283939876186880159732846332867701937330961715882746718394428627914481009391257007785408536971203165750666758077852384490279260184029373211737264310068845597273991782305778295187471341711226626762421910644338164121042775094463557363347706110614218633830535086184,0.9562940079523540929071943411882448455804659198042339938863014041593801060467576090157496959748683171680339080646917869452781885502668216669916185990922330667568398963038228431738928191227884090004548615469303071899797403388188723954061899156655653387543351937028936674153437083087070219108726088551230849901498,2.5e-300

Finding more

./find_deep.py --list                  # known regions
./find_deep.py seahorse 250            # minibrot of size ~1e-500 near the seahorse point
./find_deep.py seahorse 100 0.2,0.7    # a different minibrot at the same depth
./find_deep.py -0.1011,0.9563,38,4 80  # any Misiurewicz point, given as re,im,k,p

A disk of radius 10⁻ᴰ around a Misiurewicz point holds a minibrot of size about 10⁻²ᴰ. Newton sometimes fails to converge (elephant at depth 100 with the default offset did). The script detects this from the size and asks for a different offset. Needs mpmath.