RIVISTA DI DISEGNO
UNIVERSITÀ DEGLI
STUDI DI FIRENZE
VOL. 2 | N. 3 | 2025
MODELLI, FORME E GEOMETRIE
MODELS, SHAPES AND GEOMETRIES
72
REFRAMING DESCRIPTIVE GEOMETRY
IN THE DIGITAL ERA
The ela ionship be ween desc ip i e geome y and a chi ec u e is unde going a sig-
ni ican ans o ma ion, d i en by bo h he inc easing complexi y o con empo a y a -
chi ec u al demands and he e olu ion o digi al ools. Each echnological e a – om
adi ional desc ip i e geome y h ough 2D CAD, 3D modelling, compu a ional design,
o eme ging AI app oaches – has undamen ally al e ed he way geome y is accessed,
concep ualised, and manipula ed. These ans o ma ions ha e led o he de elopmen o
new cogni i e amewo ks o spa ial hinking. This a icle examines his ans o ma ion
h ough wo complemen a y analyses. Fi s , i in es iga es how di e en echnological
e as ha e ede ined he medium h ough which a chi ec s engage wi h geome y, o-
cusing on how hese shi s in medium ha e ans o med cogni i e mechanisms om
p ojec ion-based easoning o algo i hm-based hinking and, mo e ecen ly, o na u-
al language in e ac ion. Second, he a icle iden i ies desc ip i e geome y's e ol ing
oles in con empo a y a chi ec u al p ac ice, esea ch, and educa ion, e ealing bo h
in a ian p inciples ha pe sis ega dless o echnological mediums and new geome ic
compe encies equi ed by con empo a y a chi ec u al challenges. The analysis demon-
s a es ha desc ip i e geome y ep esen s no a ixed ope a ional me hodology bu
an e ol ing amewo k o spa ial easoning ha anscends speci ic echnological im-
plemen a ions. This con ibu es o he ongoing dialogue abou geome ic li e acy in he
compu a ional age.
Keywo ds: Desc ip i e Geome y, A chi ec u e, G aphic Exp ession, Geome ic Thinking,
Digi al Tools.
Ci a ion: J.A. Ba e a, R. Na áez Rod íguez, Re a-
ming Desc ip i e Geome y in he Digi al E a, in
TRIBELON, II, 2025, 3, pp. 72-79.
ISSN (s ampa): 3035-143X
ISSN (online): 3035-1421
doi: h ps://doi.o g/10.36253/ ibelon-3355
Recei ed: Ma ch, 2025
Accep ed: Ap il, 2025
Published: June, 2025
Copy igh : 2025 Ba e a J.A., Na áez Rod íguez
R., his is an open access pee - e iewed a icle
published by Fi enze Uni e si y P ess (h p://www.
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iles.
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JOSè ANTONIO bARRERA, RObERTO NARVáEz RODRíGuEz
Desc ip i e Geome y a he
C oss oads o A chi ec u al E olu ion
A chi ec u e p ac i ione s had been em-
ploying p ojec ion echniques o sol e
spa ial p oblems long be o e Monge’s
sys ema iza ion1. The his o ical e olu-
ion o hese me hods e lec s a chi ec-
u e’s con inuous nego ia ion wi h a ail-
able media, whe e desc ip i e geome y
me hods e ol ed in esponse o chang-
ing concep ualiza ions o space ac oss
a chi ec u al his o y2. Desc ip i e geom-
e y hus ep esen s a con inuous ajec-
o y o geome ic hinking ha ex ends
beyond Monge’s o maliza ion3. Monge
himsel acknowledged his his o ical
con inui y while a icula ing he disci-
pline’s dual objec i e: o es ablish me h-
ods o ep esen ing h ee-dimensional
bodies on wo-dimensional su aces and
o deduce spa ial u hs om hese exac
cons uc ions4 . This o mula ion e eals
wo pe asi e ea u es o any geome ic
ool ha signi ican ly in luence a chi ec-
u al p ac ice: he ep esen a ional medi-
um (pape in desc ip i e geome y) and
he me hods de eloped speci ically o
o e come he inhe en limi a ions o ha
medium. This in e dependence be ween
medium and me hod cons i u es a un-
damen al dynamic in he e olu ion o a -
chi ec u al ep esen a ion and hinking.
This ela ionship no only highligh s he
adap abili y o geome ic ools bu also
emphasizes hei ole in shaping he he-
o e ical and p ac ical amewo ks o a -
chi ec u al design. Con empo a y a chi-
ec u e aces challenges o unp eceden -
ed complexi y – including sus ainabili y
impe a i es, u ban me abolism op imi-
za ion, pe o mance-based design e-
qui emen s, and digi al ab ica ion – ha
ede ine he geome ic p oblems a chi-
ec s mus add ess5. The echnological
pa adigm has shi ed om ep esen a-
ional sys ems a emp ing o o e come
wo-dimensional limi a ions o compu-
1 Ta on, L’œu e scien i ique de Monge.
2 E ans, The p ojec i e cas : a chi ec u e and i s
h ee geome ies; Pé ez-gómez, Ques ions o
ep esen a ion: he poe ic o igin o a chi ec u e,
pp. 217-225.
3 Miglia i, Desc ip i e Geome y: F om i s Pas o
i s Fu u e, pp. 555-571.
4 Monge, Géomé ie Desc ip i e - Lecons données
aux Écoles no males, l’An 3 de la République.
5 C . Ca l, U ban Densi y and Block Me abolism, p.
852; Hensel, Da a-d i en design o A chi ec u e
and En i onmen In eg a ion Con e gence o
da a-in eg a ed wo k lows o unde s anding
and designing en i onmen s.
73
Re aming Desc ip i e Geome y in he Digi al E a J. A. Ba e a, R. Na áez Rod íguez
a ional en i onmen s whe e geome y
is di ec ly manipula ed, analysed, and
op imised in a i ual h ee-dimensional
space6. This ans o ma ion aises c i i-
cal ques ions abou he con empo a y
ele ance o desc ip i e geome y, as i s
ounda ional p emise has been echno-
logically ede ined7 ( ig. 1). This a icle
examines desc ip i e geome y’s ans-
o ma ion ac oss echnological e as,
analysing how di e en media ede ine
bo h geome ic ools and a chi ec u al
hinking. The e olu ion om 2D d a -
ing h ough 3D modelling o algo i hmic
design and eme ging AI applica ions
has p og essi ely abs ac ed geome y
om di ec manipula ion, c ea ing new
concep ual amewo ks o spa ial hink-
ing8. Th ough examina ion o scien i ic
li e a u e and con empo a y p ac ice, we
iden i y bo h pe sis en p inciples ha
anscend echnological shi s and he
new geome ic compe encies equi ed
by cu en a chi ec u al demands.
The E olu ion o he Medium:
How Technology Reshapes Geome ic
Tools and Thinking
The medium h ough which a chi ec s
engage wi h geome y plays a unda-
men al ole in shaping bo h concep ual
amewo ks and design me hodologies.
Each echnological ansi ion – om
manual d a ing o compu a ional de-
sign and, mo e ecen ly, AI-assis ed
p ocesses – econ igu es he cogni i e,
ope a ional, and epis emological dimen-
sions o geome ic p ac ice. This sec ion
examines hese echnological ansi ions
no me ely as ins umen al de elop-
men s bu as pa adigma ic shi s ha
eshape how a chi ec s concep ualize,
ope a e, and e en manage geome ic
in o ma ion. The analysis o eg ounds
he ecip ocal ela ionship be ween ep-
esen a ional media and a chi ec u al
hinking, demons a ing how each ech-
nological medium simul aneously ena-
bles new geome ic possibili ies while
imposing speci ic cons ain s and con-
cep ual amewo ks ha shape a chi ec-
u al cogni ion and p ac ice9.
P e-Digi al E a: P ojec ion as Concep ual
F amewo k
The p e-digi al e a o a chi ec u al ge-
ome y was cha ac e ised by a unda-
men al cogni i e amewo k p edica ed
on p ojec ion, bo h as a echnical p oce-
du e and as a concep ual appa a us. This
p ojec ion-based concep ual amewo k
necessi a ed sophis ica ed men al ope -
a ions o spa ial isualiza ion, equi ing
a chi ec s o engage in mul iple cogni-
i e p ocesses o a con inuous cycle o
men al codi ica ion and decodi ica ion
be ween wo-dimensional ep esen a-
ions and h ee-dimensional spa ial con-
cep s o each geome ic ope a ion10.
This ansla ion p ocess’s cogni i e load
shaped a chi ec u al p ac ice h ough
analy ical decomposi ion and sequen-
ial easoning11, es ablishing an epis e-
mological dis ance be ween cogni i e
p ocesses and a chi ec u al objec s12.
A chi ec s necessa ily ope a ed in a e-
lec i e con e sa ion h ough media ed
ep esen a ional sys ems, c ea ing ge-
ome ic easoning modes bound o p o-
jec ion-based echniques’ cons ain s
and a o dances13.
2D CAD E a: The i s digi al ansi ion
The ansi ion o 2D CAD sys ems ep-
esen ed a nuanced echnological e o-
lu ion in a chi ec u al ep esen a ion.
While undamen ally p ese ing he
cogni i e p ocess o ansla ing be ween
wo-dimensional and h ee-dimensional
concep ions, hese sys ems in oduced
c i ical ad ancemen s. Ea ly CAD imple-
men a ions p ima ily unc ioned as dig-
i al analog o adi ional d a ing p ac-
6 Johnson, Ske chpad III. Th ee-Dimensional
G aphical Communica ion wi h a Digi al
Compu e .
7 Ca po, The Digi al Tu n in A chi ec u e 1992-
2012.
8 Ca po, The Second Digi al Tu n: Design beyond
In elligence; Menges & Ahlquis , Compu a ional
Design Thinking: Compu a ion Design Thinking.
9 C . Oxman, Theo y and design in he i s digi al
age, pp. 229-265.
10 Akin e al., P oblem s uc u ing in a chi ec u al
design.
11 Akin, Psychology o A chi ec u al Design.
12 Oxman, The hinking eye: isual e-cogni ion in
design eme gence, pp. 135-164
13 E ans, The p ojec i e cas : a chi ec u e and i s
h ee geome ies, ci .
1 | A chi ec u al needs, medium challenges, and
me hods. Con as s be ween con en ional a chi-
ec u e (Desc ip i e Geome y) e sus con empo a y
a chi ec u e (Digi al Tools).
Desc ip i e
geome y’s
ans o ma ion ac oss
echnological e as,
analysing how di e en
media ede ine bo h
geome ic ools and
a chi ec u al hinking.
“
Desc ip i e
Geome y
Digi al
Tools
con en ional
a chi ec u e
- S anda dized Fo ms
- Euclidean Geome y
- S a ic Rep esen a ion
-G aphic in o ma ion
- ...
Rep esen ing
3D objec s on
2D su ace
- O hog aphic P ojec ion
- Geome ic Cons uc ions
- Manual Codi ica ion
-...
con empo a y
a chi ec u e
a chi ec u al needs medium challenge me hods
- Complex Geome ies
- Complex Rela ions
- Dynamic Rep esen a ion
- Simula ion and Op imiz.
- Da a In eg a ion
- .....
Managing complex
in o ma ion lows
- 3D isualiza ion
- Digi al Modelling
- Algo i hmic De ini ion
- AI and ML sys ems
- Da a In eg a ion
- ...
COMPLEX VIRT. 3D ENVIRONMENT COMPUTATION
PAPER
SIMPLE DESCRIPTIVE GEOMETRY
74
03/25 TRIBELON Modelli, o me e geome ie
ices, eplica ing es ablished ep esen a-
ional con en ions14, and main aining he
exis ing epis emological dis ance while
al e ing hei ma e ial subs a e.
C i ically, 2D CAD sys ems signi ican ly
enhanced geome ic p ac ice h ough
inc eased p ecision, p oduc i i y and
manipulabili y, allowing new cons uc-
ions beyond s aigh edge and compass
limi a ions15. These capabili ies sub ly e-
con igu ed ep esen a ion and concep-
ion, expanding geome ic possibili ies
while p ese ing undamen al p ojec-
ion-based cogni i e amewo ks16.
3D Modeling E a: Libe a ion om
p ojec ion as a concep ual de ice
The eme gence o 3D modelling sys ems
ma ked a undamen al epis emological
shi in a chi ec u al geome y, adically
ans o ming he ela ionship be ween
concep ion and ep esen a ion. Unlike
p e ious pa adigms ha equi ed con-
s uc ing h ee-dimensional unde s and-
ing om wo-dimensional p ojec ions,
3D modelling es ablished a digi al en i-
onmen o di ec spa ial manipula ion o
geome y, elimina ing he adi ional cod-
i ica ion-decodi ica ion cycle, in oducing
a new cogni i e amewo k and educing
he epis emological dis ance be ween
concep ion and ep esen a ion17.
This app oach acili a ed no el explo-
a ions o geome ic complexi y18, and
p o ided a con inuous spa ial easoning
app oach ha undamen ally al e ed a -
chi ec u al hinking19. Building In o ma-
ion Modeling (BIM) u he expanded
his ans o ma ion, con e ing geome -
ic models om me e ep esen a ions o
comp ehensi e in o ma ion sys ems ha
connec concep ualiza ion, analysis, and
ab ica ion h ough in eg a ed digi al
s uc u es20.
Compu a ional Design E a:
F om objec s o p ocesses
The Compu a ional Design E a ma ks a
undamen al shi om he di ec ma-
nipula ion o geome ic objec s in 3D i -
ual en i onmen s (compu e iza ion) o
he au oma ion o design asks h ough
algo i hmic p ocesses (compu a ion)21.
A chi ec s now a icula e ule-based sys-
ems ha gene a e geome y a he han
di ec ly manipula ing o m.
This ansi ion es ablishes algo i hms as
a new cogni i e medium, in oducing
a ep esen a ional language oo ed in
p og amming22. This shi demands an
unp eceden ed le el o abs ac ion and
a ionaliza ion in a chi ec u al hinking.
Designe s mus engage wi h geome y
h ough mul iple laye s: om di ec ma-
nipula ion o pa ame ic ela ionships
o algo i hmic p ocesses23. The p ac ice
ans o ms om isually-media ed o one
inc easingly go e ned by ma hema ical
and algo i hmic o mula ions24 ( ig. 2).
This ans o ma ion es ablishes a cog-
ni i e amewo k o p ocedu al hink-
ing and ma hema ical o malism. Pa -
adoxically, while educing he dis ance
be ween concep ion and geome ic
ou come – bo h i ual and physical – i
inc eases he epis emological dis ance
h ough g ea e le els o abs ac ion in
accessing geome ic po en iali ies25.
Eme ging AI E a: F om explici o implici
geome ic de ini ion
The in eg a ion o a i icial in elligence
in o a chi ec u al design ma ks ano he
undamen al epis emological ans o -
ma ion in geome ic ope a ion. This pa -
adigm shi econ igu es he cogni i e
medium h ough which geome y is ac-
cessed, ansi ioning om explici algo-
i hmic de ini ion o implici gene a ion
h ough machine lea ning sys ems.
AI in oduces a na u al language in e -
ace o geome ic manipula ion26, al-
lowing communica ion wi h geome y
h ough desc ip i e e ms a he han
o mal ma hema ical a icula ions in an
abs ac p og amming language.
14 Bha nani e al., CAD usage in an a chi ec u al
o ice: F om obse a ions o ac i e assis ance,
pp. 243-255.
15 Ba e a-Ve a, Elicon.
16 Eas man, A chi ec u al CAD: a en yea
assessmen o he s a e o he a , pp. 289-292.
17 Kola e ic, A chi ec u e in he Digi al Age: Design
and Manu ac u ing.
18 Aish & Noakes, A chi ec u e wi hou numbe s–
CAAD based on a 3D modelling sys em, pp. 321-
328.
19 Woodbu y, S a egies o In e ac i e Design
Sys ems.
20 Eas man, Building P oduc Models: Compu e
En i onmen s, Suppo ing Design and
Cons uc ion.
21 C . Te zidis, Algo i hmic A chi ec u e.
22 Aish, DesignSc ip : Scalable Tools o Design
Compu a ion, pp. 87-95; Aish, Fi s Build You
Tools, pp. 39-49.
23 Woodbu y, Elemen s o Pa ame ic Design.
24 Menges & Ahlquis , Compu a ional Design
Thinking: Compu a ion Design Thinking, ci .
25 Oxman, Thinking di e ence: Theo ies and
models o pa ame ic design hinking.
2 | Cogni i e laye s ac oss echnological e as: how
a chi ec s access and manipula e geome ic objec s.
Mo e laye s ep esen a g ea e epis emological di-
s ance be ween he a chi ec 's mind and geome y,
wi h he 3D modelling e a p o iding he mos di ec
access.
P ojec ion
decoding
(Men al spa ial
econs uc ion)
P ojec ion
decoding
(Men al spa ial
econs uc ion)
Di ec
geome ic
manipula ion
Abs ac
geome ic
manipula ion
Abs ac
geome ic
manipula ion
P ojec ion
encoding
P ojec ion
encoding
Concep uali-
za ion
o Geome ic
Logic
P ocedu al
Abs ac ion Algo i hmic
Fo mula ion
Syn ac ic
Implemen a ion
Na u al
Language
Desc ip ion
I e a i e
Re inemen
P e-digi al
2D-CAD
3D-CAD
Compu a ion
AI
E as Cogni i e laye s
Ou pu
In e p e a ion
75
Re aming Desc ip i e Geome y in he Digi al E a J. A. Ba e a, R. Na áez Rod íguez
This ans o ma ion al e s he epis emo-
logical s uc u e o geome ic p ac ice
by posi ioning machine cogni ion as an
in e p e i e in e media y be ween de-
signe in en ion and geome ic mani-
es a ion, es ablishing a collabo a ion
be ween human in en and machine in-
e p e a ion27.
AI sys ems again educe he epis emo-
logical dis ance h ough in ui i e in e -
aces28, Geome ic knowledge mus now
be e o mula ed h ough linguis ic s a -
egies a he han di ec geome ic spec-
i ica ion. This econ igu a ion ans o ms
a chi ec u al geome ic cogni ion in o a
dis ibu ed cogni i e sys em whe e ex-
pe ise is nego ia ed be ween human
and machine in elligence h ough i e -
a i e p ocesses o p omp o mula ion,
gene a ion, and e alua ion29
ing uniqueaspec s o desc ip i e geom-
e y’s e ol ing ele ance.
By analysing
desc ip i e geome y’s ela ionship o
con empo a y a chi ec u al challenges
h ough his ipa i e s uc u e, we can
iden i y bo h he endu ing concep u-
al ounda ions ha anscend pa icula
echnological implemen a ions and he
eme gen geome ic compe encies ne-
cessi a ed by con empo a y a chi ec u al
demands.
In A chi ec u al P ac ice: F om di ec
applica ion o concep ual ounda ion
The di ec applica ion o adi ional de-
sc ip i e geome y me hods has o -
en become obsole e in p o essional
wo k lows, p ima ily due o inhe en
limi a ions in i s media ing appa a us.
Examining Monge’s o iginal objec-
i es – ep esen a ion and analysis o
h ee-dimensional o ms – e eals sig-
ni ican sho comings when applied o
con empo a y a chi ec u al demands.
The ep esen a ional unc ion o de-
sc ip i e geome y has been e ec i ely
supe seded by h ee-dimensional digi al
en i onmen s ha elimina e he need
o codi ica ion-decodi ica ion p ocesses
be ween dimensions. Fo ins ance, Build-
ing In o ma ion Modeling amewo ks
es ablish geome ic consis ency ac oss
ep esen a ions while simul aneously in-
eg a ing non-geome ic da a a ibu es
ha adi ional p ojec i e me hods can-
no accommoda e30.
26 Li e al., Gene a i e AI o A chi ec u al Design: A
Li e a u e Re iew, pp. 1-32.
27 Webe e al., Designing success ul Human-AI
Collabo a ion o C ea i e-P oblem Sol ing in
A chi ec u al Design.
28 Choi e al., Gene a i e a chi ec u al plan
d awings o ea ly design decisions: da a
g ounding and addi ional aining o speci ic
use cases.
29 Shuyao e al., Towa ds Human-AI Collabo a i e
A chi ec u al Concep Design ia Seman ic AI,
pp. 68-82.
3 | Gene al design and isualiza ion app oach and
gene a i e app oach acco ding o Dai Shuyao.
(a) Gene al Design & Visualiza ion App oach
Design Concep
Design
Requi eme s
D awing Modeling Rende ing
Design Tools
INPUT OUTPUT
PROCESSING
Plan D awing
S akeholde s' Ea ly
Design Re iew
Gene a i e A chi ec u al
Plan D awings
Use Choice
Use
(in cha ge o Design) (b) P oposed App oach
P omp Addi ional T ained
Gene a i e Model
Image
The New Roles o Desc ip i e Geome y
in Con empo a y A chi ec u e
The p e ious pe sis en nego ia ion
be ween echnological dis up ion and
disciplina y con inui y es ablishes a
heo e ical amewo k o examining
desc ip i e geome y’s con empo a y
oles. This analysis necessi a es di e -
en ia ion be ween h ee in e connec ed
domains; p ac ice, esea ch, and educa-
ion, as each engages wi h geome ic
p inciples h ough dis inc epis emolog-
ical amewo ks, ope a ional con ex s,
and de elopmen al ajec o ies, e eal-
76
03/25 TRIBELON Modelli, o me e geome ie
Simila ly, he analy ical unc ion, de-
ducing spa ial u hs h ough geome -
ic cons uc ion, has been ans o med
h ough compu a ional analysis ools
ha p o ide quan i a i e assessmen
ac oss mul iple pe o mance c i e ia si-
mul aneously31. Despi e his echnolog-
ical supe session, undamen al geome -
ic easoning pe sis s as an essen ial
concep ual ounda ion wi hin compu a-
ional wo k lows. T adi ional geome ic
ope a ions ha e been ans o med a h-
e han abandoned; in e sec ions, de-
elopmen s, and p ojec ion ope a ions
emain undamen al componen s wi hin
compu a ional p ocesses, albei econ-
cep ualised h ough digi al p ocesses32.
This ans o ma ion ep esen s an ope -
a ional con inui y h ough echnological
dis up ion, whe e es ablished geome ic
li e acy emains essen ial while i s im-
plemen ing mechanisms a e adically
econ igu ed33 e ol ing owa ds a com-
pu a ional geome ic easoning whe e
adi ional p inciples in o m algo i hmic
p ocesses wi hin digi al en i onmen s34.
Howe e , p ojec ion-based ep esen a-
ions main ain limi ed ye signi ican
oles in con empo a y p ac ice, p ima -
ily in wo con ex s: (1) documen a ion
o exis ing condi ions whe e his o ical
plans cons i u e p ima y in o ma ion
sou ces, pa icula ly in he i age- ela ed
p ojec s35; and (2) egula o y compli-
ance p ocesses equi ing s anda dised
o hog aphic documen a ion. These
con ex s necessi a e bidi ec ional ex-
change be ween digi al sys ems and
p ojec ion-based ep esen a ions, c e-
a ing hyb id geome ic wo k lows whe e
p ac i ione s mus sys ema ically ans-
la e be ween compu a ional models and
con en ional p ojec ion me hods o na -
iga e p ac ical, legal, and his o ical con-
s ain s36.
30 Schia i e al., BIM da a low a chi ec u e wi h
AR/VR echnologies: Use cases in a chi ec u e,
enginee ing and cons uc ion.
31 Oxman, Digi al a chi ec u e as a challenge o
design pedagogy: heo y, knowledge, models
and medium, pp. 99-120.
32 Po mann e al., A chi ec u al geome y, pp.
145-164.
33 Kola e ic, A chi ec u e in he Digi al Age: Design
and Manu ac u ing, ci .
34 Pe e s, Compu a ion wo ks: The building o
algo i hmic hough .
35 Ma inez-Moya, Me odología de ecupe ación
g á ica de las po adas del Palacio Condal de
Oli a.
36 Ke nighan, Ex ac ing geome ic in o ma ion
om a chi ec u al d awings.
37 Gonzalez Quin ial, Ma in Pas o , Monge
Su aces. Gene a ion, Disc e isa ion and
Applica ion in A chi ec u e.
4 | Pho og aphs o The Ca e pilla Galle y. Bo om:
images o he algo i hmic design p ocess in G as-
shoppe o Rhinoce os 3D and he ep esen a ion,
in one single mul i iew o hog aphy p ojec ion, o
wo classic Desc ip i e Geome y heo ems applied
in he p ojec .
77
Re aming Desc ip i e Geome y in he Digi al E a J. A. Ba e a, R. Na áez Rod íguez
In A chi ec u al Resea ch:
New on ie s o geome ic explo a ion
Wi hin a chi ec u al esea ch con ex s,
desc ip i e geome y has unde gone sig-
ni ican concep ual expansion beyond i s
adi ional ope a ional applica ions. Con-
empo a y schola ship e eals a leas
h ee dis inc ajec o ies whe e desc ip i e
geome y p inciples emain ins umen al
despi e echnological ans o ma ion.
His o ical a chi ec u al esea ch ep e-
sen s he i s ajec o y, whe e esea ch-
e s in es iga ing p ima y his o ical
documen s – s e eo omy and mason y
ea ises, a chi ec u al manusc ip s, con-
s uc ion manuals, o builde s’ d awings
– mus necessa ily de elop expe ise
in pe iod-speci ic desc ip i e geome-
y me hodologies o ex ac meaning-
ul insigh s. The in e p e a ion o hese
his o ical documen s equi es ope a-
ional luency wi h p ojec ion-based
ep esen a ional sys ems ha go e ned
a chi ec u al concep ion and ma e iali-
sa ion du ing hei espec i e pe iods.
Compu a ional ein e p e a ion o clas-
sical desc ip i e geome y p inciples
cons i u es a second signi ican ajec-
o y in con empo a y esea ch. Th ough
algo i hmic e o mula ion o adi ional
heo ems and p inciples, he ollowing
examples demons a e he endu ing el-
e ance o hese p inciples wi hin com-
pu a ional design amewo ks37. The Ca -
e pilla Galle y ( ig.4), which implemen s
Monge’s heo em on quad ic su aces o
38 Na aez-Rod iguez, Ma in-Pas o , F om De-
sc ip i e Geome y o A chi ec u al Geome y:
Con ibu ions by Classic Au ho s o he New
Pa adigm.
39 C . Na áez-Rod íguez e al., Ligh weigh
Conical Componen s o Ro a ional Pa abolic
Domes: Geome ic De ini ion, S uc u al Be-
ha iou , Op imisa ion and Digi al Fab ica ion,
pp. 378-397; Ma ín-Pas o , Na aez-Rod i-
guez. New P ope ies Abou he In e sec ion o
Ro a ional Quad a ic Su aces and Thei Appli-
ca ions in A chi ec u e; Na áez-Rod íguez e
al., Ligh weigh Conical Componen s o Ro a-
ional Pa abolic Domes: Geome ic De ini ion,
S uc u al Beha iou , Op imisa ion and Digi al
Fab ica ion ci ., pp. 378-397.
40 Sch eibe , Gene alized Desc ip i e Geome y.
gene a e complex spa ial con igu a ions
h ough de elopable conical su aces38;
and The A chimedean Pa ilion ( ig. 5),
which employs a p ojec i e in e p e a-
ion o A chimedes’ heo em o enable
he disc e iza ion o pa abolic domes39.
These cases demons a e how he ole
o geome ic easoning in compu a ional
wo k lows and geome ically-in o med
algo i hms signi ican ly op imize compu-
a ional p ocesses by le e aging wo-di-
mensional p ojec i e p ope ies o de-
ine h ee-dimensional complex o ms,
he eby educing algo i hmic complexi y
while simul aneously enhancing s uc-
u al pe o mance, ma e ial e iciency,
and ab ica ion easibili y wi hin con-
empo a y a chi ec u al p oduc ion.The
hi d ajec o y, al hough ep esen ing
ela i ely isola ed occu ences wi h-
in he b oade li e a u e, demons a es
he ema kable adap abili y o desc ip-
i e geome y p inciples ac oss di e se
esea ch domains beyond con en ional
a chi ec u al applica ions. This in e dis-
ciplina y expansion includes inno a i e
“ e e se desc ip i e geome y” me hod-
ology, which econs uc s 3D mason y
s uc u es om 2D images; applica ion
o high-dimensional desc ip i e geom-
e y o neu al ne wo k pa e n ecogni-
ion sys ems; and heo e ical amewo k
o “Gene alised Desc ip i e Geome y”
(GDG) o cons uc i ely imaging ab-
s ac ma hema ical objec s h ough al-
go i hmic ansla ions40.
5 | Pho og aphs o The A chimedean Pa ilion.
78
03/25 TRIBELON Modelli, o me e geome ie
In A chi ec u al Educa ion:
New Compe ences o New Demands
Con empo a y a chi ec u al educa ion
mus nego ia e he complex in e ace be-
ween endu ing geome ic p inciples and
apidly e ol ing echnological pa adigms.
Analysis o he undamen al epis emo-
logical ans o ma ions documen ed in
p e ious sec ions could e eal h ee es-
sen ial compe ency clus e s o geome -
ic educa ion in he compu a ional e a:
- Fi s , s uden s equi e p o iciency in
C ea ing Geome ic Models, Simula ions
and Algo i hmic De ini ions in Digi al En-
i onmen s, encompassing he cons uc-
ion o basic geome ic elemen s, anal-
ysis o in e ac ions be ween hem, gen-
e a ion o composi e objec s, applica ion
o ans o ma ions, modeling o complex
su aces wi h embedded ab ica ion pa-
ame e s, and ex ac ion o geome ic
in o ma ion om complex assemblies.
This compe ency clus e adap s adi-
ional desc ip i e geome y ope a ions
o compu a ional amewo ks while
main aining concep ual con inui y.
- Second, and mos c i ically o u u e
p ac ice as he compe encies in he
i s clus e become inc easingly au o-
ma ed by a i icial in elligence sys ems,
s uden s mus de elop capabili ies o
Managing Flows o In o ma ion Be ween
Physical and Digi al En i onmen s. This
eme gen compe ency domain in ol es
unde s anding and ansla ing geome ic
in o ma ion ac oss di e se ep esen a-
ional sys ems, e alua ing ab ica ion
p ocesses, and e ec i ely communi-
ca ing geome ic in o ma ion h ough
mul iple pla o ms, including human
and a i icial in elligence in e ac ions.
This clus e ep esen s a undamen al
e olu ion beyond adi ional desc ip i e
geome y, add essing he inc easingly
pe meable bounda y be ween digi al
concep ion and physical ma e ializa ion.
- Thi d, s uden s con inue o equi e
compe ency in In e p e ing and Gene -
a ing O hog aphic P ojec ions wi h P e-
cision, hough now con ex ualised wi hin
compu a ional wo k lows. This main ains
connec ion wi h his o ical ep esen a-
ional con en ions while acknowledging
hei econ igu ed ole wi hin con em-
po a y p ac ice.
p ojec ion-based sys ems o compu a-
ional amewo ks. Con empo a y p ac-
ice demons a es ha while adi ional
applica ions ha e diminished, he un-
de lying concep ual ounda ions in o m
eme ging compu a ional me hodologies
ac oss p ac ice, esea ch, and educa ion.
This econcep ualiza ion necessi a es e-
de ining desc ip i e geome y no as a
ixed ope a ional me hodology, bu as an
e ol ing amewo k o spa ial easoning
ha anscends pa icula echnolog-
ical implemen a ions. Fu u e esea ch
should u he examine how geome ic
compe encies ans e ac oss echno-
logical ansi ions, pa icula ly as AI sys-
ems in oduce new cogni i e in e aces
o geome ic ope a ion. Ul ima ely, de-
sc ip i e geome y’s endu ing ele ance
lies no in i s his o ical echniques bu in
i s concep ual appa a us o na iga ing
he inc easingly complex ela ionship
be ween a chi ec u al concep ion and
ma e ializa ion.
6 | P ojec i e in e p e a ion o A chimedes’ p oposi-
ion o disc e ising he pa aboloid’s su ace wi h pla-
na ellip ical aces s emming om a ci cle-packing
algo i hm. On he igh : A chimedes’ p oposi ion in
A chimedes-Mau olico: De conoidibus e sphae oidi-
bus igu is In en o um. Libe secundus, P oposi ion
XII, Pale mo 1685.
Conclusion: Towa ds a New
Unde s anding o Desc ip i e
Geome y
This analysis e eals ha desc ip i e
geome y’s ans o ma ion ac oss ech-
nological e as ep esen s nei he ob-
solescence no unchanged con inui y,
bu a he a undamen al econcep ual-
iza ion o i s p inciples wi hin e ol ing
a chi ec u al pa adigms. The pe sis en
cogni i e ope a ions o desc ip i e ge-
ome y – spa ial isualiza ion, geome -
ic easoning, and analy ical decompo-
si ion – emain essen ial, e en as hei
implemen ing mechanisms e ol e om
79
Re aming Desc ip i e Geome y in he Digi al E a J. A. Ba e a, R. Na áez Rod íguez
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