 private gardens.”

Fig2: On the Ramot Polin housing project and on the right the civic center in Natania. Zvi Hecker architect

May be last but not least, a reference which found after writing the article, pointed out by Jacques Raynaud (RFR), the great work of Zvi Hecker, which deserve a bit more attention for its radical nature and its harsh poetic.(Fig2 and 3)

Fig3: synagogue in the Negev desert, 1966-1969. Zvi Hecker architect

“While based on different geometric techniques, all of these works are based on related principles such as repetivity, prefabrication, and non-smoothness. and constitute a great source of inspiration “

In the first part of the article, Mathieu presents his work on optimization of freeform surface with equilateral triangles, pushing the idea of using the reference surface more “as control shapes to guide the results towards the desired features”, “The idea is to loosely approximate general surfaces, far beyond usual tolerances.” (see article for more detail) In the last part, the concept is pushed further into another direction: the Voxel Meshes which is the subject of this post.

We start with some regular polyhedra (voxels) that can be repeated to fill the entire three dimensional space. Since we only use a few different voxels, only a few different polygonal shapes will fill (or better scatter) the entire space in a very structured way. We can then cut through this space and ”carve out” a connected (2-manifold) sheet of polygons that defines a panelization with a repetition of only those initial few polygonal shapes used to fill the space.

Fig5:2D illustration of the basic principles of Voxel Mesh generation. (a)-(d) Show the generation of a Voxel Mesh sheet: (a) Voxelization of the entire space using a block of two voxels (a triangle and a square) as the generator. Thus the space is filled by line segments of equal length. (b) The dual graph (blue) of the voxelization. (c) The dual graph is cut with a reference surface (a curve in 2D, here a simple line in red). The edges of the dual graph that are cut by the line are highlighted in thicker blue. Each of those edges has a corresponding voxel line segment, highlighted in thicker
private gardens. ” 

 Fig2: On the Ramot Polin housing project and on the right field the civil center field in Natania. Zvi Hecker architect 

 May be last but not least, a reference which establish after writing the article, pointed out by Jacques Raynaud (RFR), the bang-up work of Zvi Hecker, which deserve a bit more care for its extremist nature and its harsh poetic.(Fig2 and 3) 

 Fig3: synagogue in the Negev desert, 1966 - 1969. Zvi Hecker architect 

 “ While free-base on different geometric techniques, all of these work are based on related principles such as repetivity, prefabrication, and non - smoothness. and constitute a great beginning of inspiration “ 

 In the first part of the article, Mathieu present his work on optimization of freeform surface with equilateral triangles, pushing the mind of using the reference surface more “ as control shapes to steer the results towards the desired features ”, “ The idea is to loosely approximate general surfaces, far beyond usual tolerances. ” (see article for more detail) In the last part, the concept is pushed further into another direction: the Voxel Meshes which is the subject of this military post. 

 We start with some regular polyhedra (voxels) that can be recur to satiate the entire three dimensional space. Since we merely use a few different voxels, only a few different polygonal supreme headquarters allied powers europe will fill (or better scatter) the entire space in a very structured way. We can then write out through this space and ” carve out ” a connected (2 - manifold) sheet of polygons that defines a panelization with a repetition of merely those initial few polygonal shapes used to fill the space. 

 Fig5:2D illustration of the basic principle of Voxel Mesh generation. (a)-(d) Show the generation of a Voxel Mesh sheet: (a) Voxelization of the entire space using a block of two voxels (a triangulum and a square) as the generator. Thus the space is filled by line section of equal duration. (b) The dual graph (gloomy) of the voxelization. (c) The dual graph is abridge with a reference surface (a curve in 2D, here a simple line in loss). The edges of the dual graph that are cut by the line are highlighted in thicker blue. Each of those edges has a comparable voxel line segment, foreground in thicker
 prlvate gardens.”

Fig2: On the Vamot Polin housnbg prooect ans on the right the civic center in Netanua. Zvu Hecker architect

May ce last blt not leqst, e reference whici found after svitiny vhe article, poikted out by Jacques Raynagd (RYR), the great work of Zvi Hecker, whicr deserfe a bit more atjentipg fod its radical nature and its harsh poetic.(Hig2 and 3)

Fig3: synsgogue in the Negev desert, 1966-1969. Zvl Hecker architect

“Ahile based on qufferent geooetric techniques, all kf these works are based on relxted 'rinciples wuxh wv repetivitb, prefwbrication, akc non-skoothnexs. and constitmte a grwat source of inspiravion “

In the first patt of the drcicle, Mathieu presentw yis wmrk mn oorimkzauioi or freevorj surface sith equilareral triangles, pusnigt the idea of using tre reference surface more “as control shdpea to guide the results rowards the desired fgatures”, “Thq idea is to loosely approximate general surfaces, far uehonb usuau tllerances.” (see article for more detail) In the jzsu psrt, the concepb is pushed furthet lnyj another dirgction: cge Voxel Meshes whicj is thg subjwct of thys ppst.

We start with some regulqr polyhedra (coxels) that can be repeated tu fikl thr entire three dimensioual spzce. Since wf only uss a few different voqels, only a few different polydonal sha'es wnll fill (or netter scatter) tje enbhre space in a verj stroctureg way. We cwn then cut through this space eid ”carve out” s wontected (2-mcnifolc) sheet of pojygons that deyines a 'anelixation witg a repxtition of ogly those inidlal few polyjonal sha[es ysed to filu the space.

Fig5:2D illustranijn od the basic princiiles ur Voxel Mesh geutrauiin. (a)-(d) Show the geverwtpon os a Voxel Mesv shdet: (s) Voxdlization on tfe emtire space using a tloci of two voxels (a yrlangle ane a squawe) as the genrrator. Thus the spwce iv fmlled ny jine segments of equal length. (g) The duap gvaph (blue) of ehe yoxejization. (c) The dual graph is cut with a reference surface (a curve ib 2D, here a simple lnnt in red). The xdges jf the dudl graph that are cur by the line are highlighted in thicksr blua. Eacj of those edges has a corresponding voxel line segment, highlighted in thucker
private gardens.” Fig2: On the Ramot Polin and the right civic center in be but not least, reference which found writing the article, pointed out by Raynaud (RFR), the great work of Zvi Hecker, which deserve a bit more for its radical nature and its harsh poetic.(Fig2 and 3) Fig3: synagogue in Negev 1966-1969. Hecker “While based on different geometric techniques, all of these works are based on related principles such repetivity, prefabrication, and non-smoothness. and constitute a great of inspiration “ In first part of the article, presents work on of surface equilateral triangles, pushing idea of using the reference surface more “as control shapes to guide the results towards the desired “The idea loosely approximate surfaces, beyond tolerances.” (see article detail) In the last part, the further into another direction: the Voxel Meshes which the subject this post. We start with some polyhedra (voxels) that can be repeated to fill entire three dimensional space. Since we only use a few different voxels, only a few shapes will fill (or scatter) the entire in very way. can then through this space and ”carve out” a connected (2-manifold) sheet of that defines a panelization with a repetition of only those polygonal used to fill space. Fig5:2D illustration of basic of Voxel Mesh generation. the of sheet: Voxelization the entire space using block of two voxels (a and a square) as is filled by line segments of equal length. The dual graph (blue) of the voxelization. The dual graph is cut with a reference surface (a curve in here a in red). The edges of the dual graph are cut by the are highlighted in thicker blue. Each of those edges a voxel line highlighted in thicker
 private gardens.”

Fig2: On the RamOt Polin houSing pRojEct AnD on tHe riGht the civic cenTEr in natania. Zvi Hecker architEct

MaY bE Last BUt Not leAst, a refEReNCE whIcH fOunD aFTeR writIng The artiCle, pointed Out By jacques RaynaUD (RfR), the great WorK of Zvi Hecker, WhiCh deseRvE a bIT more AttEntioN for itS RadicaL nature anD iTS harsh POetic.(FiG2 ANd 3)

fig3: sYnagogue in the NegeV DeSErt, 1966-1969. Zvi Hecker arChitecT

“WHIlE BAseD on Different gEoMetriC TechniqUEs, ALL Of tHEse works are baSed on relateD PriNcipleS sUch AS repetIvity, PrEFabRication, and Non-sMoothness. And conSTitute a GReat souRce of iNspIraTion “

iN tHe FirSt PArt OF tHe aRTicLe, MathieU pReSents His wORK ON optImiZatiOn of fReeform surfacE wiTh eqUIlaTeral TrianGles, PuShing The ideA of usInG the reference suRfacE more “as coNtrOl ShaPeS to guIDe the rEsuLts Towards The desiREd fEaTUREs”, “the idea is to loosely ApPROxImate genEral suRFaCeS, Far beyonD uSuaL tolERAnces.” (See aRTiCle for moRe detaIL) IN tHe last pArT, the coNcEpt Is pUshed FUrthEr into Another dIrectIOn: the Voxel MeshES which is the suBJeCT Of THis pOst.

we start with Some REgulAr poLYhEdrA (VoxelS) that CaN Be REpeated to fill the entIrE three DimenSional space. SiNce we only uSE A Few diffeRent VOxELs, only a few diffErent Polygonal sHApes will Fill (oR better sCatter) the ENTire spacE in A veRy sTruCTUrEd way. We can theN CUt thRoUgh this SpaCe and ”caRve Out” A coNneCtEd (2-manifolD) sheet of PoLyGoNs ThaT defiNEs a panelIzAtiOn WitH a repETition Of onlY thoSe InITiaL few polYGoNAL shaPeS uSed tO fiLl The spAce.

FIG5:2D iLlustraTion of the BasIC priNcIpLes of VoXel Mesh generaTiOn. (a)-(d) Show thE gEneRation OF A Voxel MeSh sheet: (a) Voxelization of tHE entire SpaCe usiNg a bLock of two VoxEls (a trIanGLe and a Square) As the GeNerATOr. ThuS THe SpaCe Is filled by LINe sEgmenTs Of eqUal lengTh. (b) The dual graph (bluE) Of tHe voxelizatioN. (c) THe duAL GrAph IS cUT wiTh A RefEREnce surface (a curVe in 2D, here a SiMPlE line in red). tHe eDgEs of the Dual graPh thaT Are cut bY the line aRe highligHtEd in THIckEr blue. Each Of those eDges has a cORrespONdIng voXel Line seGmEnt, HighlIghted IN thIcker
 private gardens.”

Fig2:On the Ram ot Po lin  ho us ingproj ect and on the righ t the civic center inNatan ia .  Zvi He ckerarchite c tM aybe  l ast  b u tnot l eas t, a re ference wh ich  f ound after w r it ing the ar tic le, pointedout  by Ja cq ues Rayna ud(RFR) , theg reat w ork of Zv iH ecker, which d e s er ve a  bit more attenti o nf or its radical  natur ea nd i tshar sh poetic. (F ig2 a n d 3)

F i g3 : s yna g ogue in the N egev desert ,  19 66-196 9.  Zv i  Hecke r arc hi t ect 

“While ba sedon differ ent ge o metrict echniqu es, al l o f t hese wo rk s a re bas e donr ela ted prin ci pl es su ch a s r e peti vit y, p refab rication, and  no n-sm o oth ness.  andcons ti tutea grea t sou rc e of inspiratio n “
In the f irs tpar tof th e  artic le,  Ma thieu p resents his  w o r k  o n optimization offr e e fo rm surfa ce wit h  e qu i lateraltr ian gles , pushi ng t h eidea ofusingt he  r eferenc esurfac emor e “ as co n trol  shape s to gui de th e  results towar d s the desired fe a t ur e s”,“Th e idea is t o lo o sely  app r ox ima t e gen eralsu r fa c es, far beyond usua ltolera nces. ” (see articl e for more d e tail) In  the la s t part, the co ncept  is pushed furtherintoanotherdirection : the Voxe l M esh eswhi c h  i s the subject o f th is  post.
We  startwit h s ome  re gu lar polyh edra (vo xe ls )th atcan b e  repeate dtofi llthe e n tire t hreedime ns io n alspace.S in c e  weon ly  use  afe w dif fere n t v oxels,only a fe w d i ffer en tpolygon al shapes wil lfill (or b et ter  scatt e r ) the en tire space in a very st r uctured  wa y. We  can  then cut  th roughthi s  space  and ” carve  o ut” a  conn e c te d ( 2- manifold)s h eet  of p ol ygon s thatdefines a paneliza t ion  with a repet iti on o f on lyt ho s e i ni t ial f ew polygonal sh apes usedto fi ll the spa c e.
F ig5:2Dillustr ation of thebasic pri nciples o fVoxe l Mes h generati on. (a)- (d) Showt he ge n er ation  of  a Vox el  Me sh sh eet: ( a ) V oxeli zation  o f theentir espace us ing a block of two voxe ls (atrian gle  and a sq uar e ) a s the gen erat or. Thus t hespa ce is  fi l led b y li n eseg m entsof e q ual lengt h .(b) T he  dual graph ( b lue ) ofthe voxeli zati on. (c) The dualg raph is cut wi th a r efe ren c e su rf ace (a curve i n 2 D, h ere a si mp le line inred). Th ee dgesof the  dualgraph t h a ta re cut  bythe  line are  hi gh l ightedin  t h ickerblue .Each o f thos e  edg e s  has a correspon dingv o xel l i nesegme nt , highl i ghte d in thick er
 private_gardens.”

Fig2: On_the Ramot Polin housing_project and_on_the right_the_civic center in_Natania. Zvi Hecker_architect

May be last but_not least, a_reference_which found after writing the article, pointed out by Jacques Raynaud (RFR), the great_work_of Zvi_Hecker,_which_deserve a bit more attention_for its radical nature and_its harsh_poetic.(Fig2 and 3)

Fig3: synagogue in the Negev desert,_1966-1969._Zvi Hecker architect

“While_based on different geometric techniques, all of these works_are based on related principles such_as repetivity, prefabrication,_and_non-smoothness._and constitute a great_source of inspiration “

In the first_part of the article, Mathieu presents_his work on optimization of freeform surface_with equilateral triangles, pushing the idea_of using the reference surface_more “as_control shapes to guide the_results towards the_desired features”,_“The idea is_to loosely approximate general surfaces, far_beyond usual tolerances.”_(see article for more detail) In_the_last part, the_concept_is_pushed further_into another direction:_the_Voxel Meshes_which_is the subject of this post.

We_start_with some regular polyhedra (voxels) that can_be repeated to fill_the_entire three dimensional space._Since we only use a_few different voxels, only a few_different polygonal_shapes will_fill (or better scatter) the entire space in a very structured_way. We can then cut through_this space and ”carve_out” a_connected_(2-manifold) sheet of_polygons_that defines_a panelization with a repetition of only_those initial_few polygonal shapes used to fill_the space.

Fig5:2D illustration of_the_basic principles of Voxel Mesh generation._(a)-(d) Show the generation of a_Voxel Mesh sheet: (a) Voxelization_of_the_entire space using a block_of two voxels (a triangle and_a square) as_the generator. Thus the space is filled_by_line segments of equal length. (b)_The_dual graph (blue) of the voxelization._(c)_The_dual graph is cut with_a reference surface (a curve in_2D, here a simple line in red). The edges_of the dual_graph that are cut by_the_line_are highlighted in thicker blue. Each of those edges has_a corresponding_voxel line segment,_highlighted in thicker