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DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

Request for Contract Update

  • R142201
  • Office Furniture and Installation

Pursuant to the terms of contract number________________ for _________________________ Contractor must notify and receive approval from Region 4 ESC when there is an update in the contract. No request will be officially approved without the prior authorization of Region 4 ESC. Region 4 ESC reserves the right to accept or reject any request.

Allsteel Inc.

hereby provides notice of the following update on

( Contractor)

April 17, 2019

  • this date
  • .

Instructions: Contractor must check all that may apply and shall provide supporting documentation. Requestsreceived without supporting documentation will be returned. This form is not intended for use if there is a material change in operations, such as assignment, bankruptcy, change of ownership, merger, etc. Material changes must be submitted on a “Notice of Material Change to Vendor Contract” form.

Authorized Distributors/Dealers

Addition

Price Update

Supporting Documentation
Deletion Supporting Documentation

Discontinued Products/Services

Supporting Documentation

X

Products/Services

New Addition

X

Update Only

X

Supporting Documentation

Other

States/Territories

Supporting Documentation
Supporting Documentation

Notes: Contractor may include other notes regarding the contract update here: (attach another page if necessary).

Attached is Allsteel's request to add the following new product lines: Park, Recharge, and Townhall

Collection along with the respective price lists and updated discount matrix.

4/30/2019
X

Submitted By: Cindy Hermann

Government Financial Analyst

Approved by Email: Date Denied by Email: Date

Title:

Email Address: [email protected]

Region 4 ESC:

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

April 17, 2019 ATTN: Robert Zingelmann Region 4 Education Service Center 7145 West Tidwell Road Houston, TX 77092

RE: Request to add new products Dear Robert: Please accept this letter as Allsteel’s request to add our new Park, Recharge, and Townhall Collection product lines to our OMNIA Partners Region 4 Furniture and Installation Contract #R142201 as follows:

•

Park: Park is a collection of soft seating, tables, and accessories that blends high-quality American steel production with the meticulous beauty of Danish craft. Park is a perfectly balanced, contemporary workplace solution that preserves the usability of shared, open spaces.

••

Recharge: Recharge includes soft seating pieces and tables that can be integrated for private, public, or collaborative settings.

Townhall Collection: Townhall groups movable soft seating, tables, and accessories into multifunctional and agile micro-environments for individual and group work in shared, open spaces.

Enclosed is a discount matrix for your reference. Thank you for your continued business and opportunity to provide OMNIA Partners Region 4 end-users with quality Allsteel products. If you have any questions, please do not hesitate to contact me at

563-272-4441 or via email at [email protected].

Sincerely, Cindy Hermann Government Financial Analyst

Enclosures

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

Updated 4/17/19

OMNIA Partners Region 4 Education Service Center Discount Matrix
Allsteel Inc. Contract #R142201-Furniture and Installation

Discount From List

  • Products
  • List per Order

  • $1 - $100,000
  • 69.5%

71.0%

Stride Systems, Terrace, Concensys, Optimize, Further, Involve Casegoods, Approach, Stride Desking, Stride
Benching, Pedestals, Overheads, Worksurfaces,
Electrical & Accessories, Altitude Tables

$100,001 - $400,000

  • $400,001 & Above
  • Negotiable

Lateral Files, Vertical Files, Storage (Towers,
Bookcases, & Cabinets), Align Laterals, Align Storage,
Align Credenzas, Stride Storage, Involve Storage,
Radii Storage

$1 - $100,000
$100,001 - $400,000
$400,001 & Above
64.0% 65.0% Negotiable

$1 - $100,000
$100,001 - $400,000
$400,001 & Above
64.0% 64.0%

Trooper, Tolleson, and Nimble Seating

Negotiable
$1 - $100,000
$100,001 - $400,000
$400,001 & Above
61.0% 61.0%

#19, Relate, Sum, Scout, Seek, Inspire, and
Access Seating

Negotiable
$1 - $100,000
$100,001 - $400,000
$400,001 & Above
58.0% 58.0%

Acuity, Mimeo, Lyric, Quip, Evo, and Svelte Seating
Merge and Aware Tables

Negotiable
$1 - $100,000
$100,001 - $400,000
$400,001 & Above
64.0% 64.0% Negotiable

$1 - $100,000
$100,001 - $400,000
$400,001 & Above
60.0% 60.0%

Wood Casegoods (Align Wood), Stride Painted Wood
Casegoods Veneer

Negotiable
$1 - $100,000
$100,001 - $400,000
$400,001 & Above
51.0% 54.0%

Gather Collection, Clarity, Co|Ho, Park, Recharge, Retreat, Townhall Collection (Rock, Wedge, Peak, Summit, Picnic, Cloud), and New Product Offering

Negotiable
$1 - $100,000
$100,001 - $400,000
$400,001 & Above
55.0% 57.0%

Accessories (Wand & Link Lights, Ergonomic tools)

Negotiable

  • $1 - $200,000
  • 55.3%**

Beyond*

  • $200,001 & Above
  • Negotiable

*Authorized Beyond dealers only. Please contact [email protected] for additional information. See below for Approved Beyond Services.

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

Approved Furniture Services:

  • Fee
  • Type
  • Rate

NTE 25% net product total/order; a minimum charge of $175.00 per

Install Design
Negotiated Negotiated

order may apply.

NTE $90/hr

  • NTE $90/hr
  • Project Management Negotiated

Reconfiguration Storage
Negotiated Negotiated
NTE $125/hr
NTE $5/sq ft/month
(NTE = "Not to exceed")

Approved Beyond Services:

  • Fee
  • Type
  • Rate

NTE exceed $130/lineal foot***
NTE $90/hr

  • Install
  • Negotiated

Negotiated Negotiated Negotiated
Design Project Management Reconfiguration
NTE $90/hr NTE $125/hr
(NTE = "Not to exceed")
** Due to the extremely custom nature of demountable walls, list prices are part of the CET Designer software.
*** Installation costs noted are for normal work hours, non-union wages. Union, prevailing wage, overtime, weekend work will be negotiated on a case-by-case basis. **** Prices for Allsteel Beyond Architectural Walls Product do not include applicable sales taxes or freight. These charges will be listed as separate line items on the quote, purchase order, and invoice, unless otherwise agreed to in writing, and each charge is contingent upon final destination of product. All applicable taxes and freight charges will be added to Allsteel's invoice, and customer agrees to pay the same.

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

™

Park

BY NORM ARCHITECTS

SPECIFICATION & PRICE LIST 2019

prices effective March 28

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

Park™ Seating, Tables and Pedestals

Laminates and Edge Finishes

  • LAMINATES
  • EDGES
  • PAINT

GRADE L1-ABSTRACT

Amber Cherry ......................................DA Beigewood............................................DE Black.....................................................E4 Branded Oak........................................BO Brazilwood...........................................DB Brownstone.......................................... EY Bungalow.............................................EX Cafelle................................................. DG Carob....................................................E8 Charcoal Walnut..................................DC Columbian Walnut...............................DZ Designer White....................................DW Ebony Recon........................................DR Fawn Cypress....................................... FC Flint ......................................................E9 Florence Walnut.................................. FW Fossil....................................................EH Kingswood Walnut................................ KI Loft...................................................... EV Lowell Ash...........................................DL Mangalore Mango .............................MM Muslin.................................................. EU Natural Maple .................................... DD Natural Recon..................................... NR Natural Walnut................................... DN NeoWalnut.......................................... NU Parchment............................................EN Phantom Charcoal ............................... PC Phantom Ecru .......................................PE Platinum Metallic..................................EP Plyband.................................................PL Portico Teak......................................... DP Shaker Cherry ...................................... DF Skyline Walnut.....................................SW Titanium ............................................... TI

GRADE P1

Ash....................................................LT3B Camelstone ...................................... LT6A Canyon Zephyr.................................LT5B Desert Zephyr .................................. LT5A Grey Mesh ....................................... LT8A Greystone........................................TM65 Parchment.......................................TM20 Pewter Mesh ....................................LT8C Sheer Mesh........................................LT8B Silver Mesh ......................................LT8D White Tigris ..................................... LT2A Whitestone......................................TM33
Designer White Texture...................... PK7 Textured Flint .................................... P7A Textured Loft......................................P7L Textured Titanium ............................. P8V

GRADE P2

Silver Texture..................................... PR8 Solar Black......................................... P8X

GRADE L1-SOLID

Brownstone..................................... LM13 Bungalow........................................ LM12 Designer White 15051.................... LDW1 Flint ................................................ LM16 Fossil Shale.......................................LEH1 Loft................................................. LM11 Muslin............................................. LM15 Titanium ............................................LTI1

GRADE L1-WOODGRAIN

Florence Walnut NEW! ................... LFW1 Kingswood Walnut NEW!.................LKI1 Natural Maple ..................................LWD NeoWalnut...................................... LNU1 Shaker Cherry .................................LW7C

GRADE L2-WOODGRAIN

Amber Cherry .................................LWAC Beigewood....................................... LWBE Branded Oak....................................LBO1 Brazilwood......................................LWBR Columbian Walnut............................LWZ Ebony Recon....................................LER1 Fawn Cypress................................... LFC1 Lowell Ash ......................................LLA1 Mangalore Mango .........................LMM1 Natural Recon ................................ LNR1 Phantom Charcoal ........................... LPC1 Phantom Ecru ..................................LPE1 Portico Teak .................................... LPT1 Skyline Walnut ............................... LSW1
NOTE: $33 upcharge on Woodgrain T‑Mold.

GRADE L3-WOODGRAIN

Cafelle...............................................LWC Charcoal Walnut........................... LWCW Natural Walnut.............................LWNW

GRADE L7-SOLID (TRACELESS)*

Black Velvet .....................................LBV1 Charcoal Velvet................................LCV1 * Park Storage is available in all grades of laminate except Grade 7.

®

All of Allsteel’s tables are level – 2 certified to the ANSI/BIFMA e3 Furniture Sustainability Standard. See www.levelcertified.org for more information.

NOTE: Refer to individual product pages for upcharges.

FSC Certified Wood at Allsteel

See page 262 of Seating, Collaboration & Tables pricer for more information.

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

Park™

Specification and Price List

TABLE OF CONTENTS

Finishes ‑ Paints, Laminates, and Veneers................................... 2 Table of Contents ....................................................................... 3 Park Single‑ and Multi‑Upholstery Seating;
Park Poly Shell Chair ....................................................... 6‑11
Park Tables.......................................................................... 12‑13 Park Plant/Storage Pedestal and Easel ...................................... 14 Park Electrical Components ..................................................... 15 Partnership Program Upholstery ......................................... 17‑19 Seating Upholstery Options...................................................... 20 Standard Program Upholstery ............................................. 22‑23

© 2019, Allsteel Inc. Allsteel is a registered trademark.

™

Park by Norm Architects Supplement – 3/2019

allsteeloffice.com | 3

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

Park™ Seating, Tables and Pedestals

Typicals

GSA Reference Page

Alone-Together Solo

  • AIRPORT LOUNGE
  • WORK/LOUNGE PINWHEEL
  • TOUCHDOWN

Traditional Meeting

HI-LO CONFERENCE
WITH MEDIA
CONFERENCE
(6 PERSON)
SMALL MEETING
(4 PERSON)

Open Social Collaborative

  • COLLABORATIVE WORK LOUNGE
  • SPACE DEFINING LOUNGE

(4 PERSON)
SPACE DEFINING LOUNGE
(2 PERSON)

Kit-of-Parts

  • SPACE DEFINING BANQUETTE
  • BACK-TO-BACK LOUNGE
  • AGILE OPEN SOCIAL MEETING

Re-deployable Space Division

END-OF-RUN

™

4 |

Park by Norm Architects Supplement – 3/2019

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

Notes

™

Park by Norm Architects Supplement – 3/2019

allsteeloffice.com | 5

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

Park™ Seating

Lounge

Pending GSA Approval

(See page 16 for COM pricing on Grades 9-22)

Upcharge
Model Number

Add options below

Base

  • Price
  • Upholstery
  • 2
  • 3
  • 4
  • 5
  • 6
  • 7
  • 8

Description

Park Single Upholstery

Park Lo Solo Lounge

  • NALS
  • $ 2000
  • Upholstery
  • $

$
29 $ 57 $ 84 $ 122 $ 158 $ 195 $ 232

Park Lo Settee Lounge

NALD NAHS NAHD

  • $ 3200
  • Upholstery

Upholstery Upholstery
45 $ 89 $ 133 $ 192 $ 250 $ 309 $ 367

Park Hi Solo Lounge

  • $ 2600
  • $
  • 41 $ 81 $ 121 $ 173 $ 227 $ 279 $ 333

Park Hi Settee Lounge

  • $ 4300
  • $
  • 58 $ 115 $ 172 $ 248 $ 324 $ 399 $ 475

How to specify —

  • 1st Option:
  • 2nd Option:

Select the Model Number from above.
Select the Base Paint Finish:
Select the Upholstery:

PK7 Designer White Texture P7A Textured Flint
See Allsteel Upholstery matrix on page 20.
P7L Textured Loft P8V Textured Titanium PR8 Silver Texture
(Grade 2 upcharge: Solo $23 / Sette $24)
P8X Solar Black
(Grade 2 upcharge: Solo $23 / Sette $24)

  • N A
  • .
  • .

Example: NALS.P7L.SCAMBLZ24

™

6 |

Park by Norm Architects Supplement – 3/2019

DocuSign Envelope ID: 0E48A6C9-4184-4355-B75C-D9F791DFB905

Park™ Seating

Lounge

Pending GSA Approval

  • Overall
  • Seat
  • Back

Model Number

  • 5
  • 5
  • 1
  • 1
  • 1
  • 3

NALS

  • 50
  • 39.4
  • 2.5
  • 31 ⁄8
  • "
  • 32"

64"

  • 27 ⁄8
  • "
  • 18 ⁄4
  • "
  • 21"
  • 19 ⁄2
  • "
  • 3"
  • 17 ⁄2
  • "
  • 15"

48"

  • 8 ⁄4
  • "
  • 107"

  • 5
  • 5
  • 1
  • 1
  • 1
  • 1
  • 3

NALD

  • 80
  • 82.2
  • 4.5
  • 31 ⁄8
  • "
  • 27 ⁄8
  • "
  • 18 ⁄4
  • "
  • 55 ⁄4
  • "
  • 19 ⁄2
  • "
  • 3"
  • 17 ⁄2
  • "
  • 8 ⁄4
  • "
  • 107"

  • 3
  • 5
  • 1
  • 1
  • 3

NAHS

  • 65
  • 55.9
  • 3.5
  • 48 ⁄8
  • "
  • 32"
  • 27 ⁄8
  • "
  • 18 ⁄4
  • "
  • 21"
  • 19 ⁄2
  • "
  • 3"
  • 32"
  • 15"
  • 8 ⁄4
  • "
  • 107"

  • 3
  • 5
  • 1
  • 1
  • 1
  • 3

NAHD

  • 105
  • 116.1
  • 5
  • 48 ⁄8
  • "
  • 64"
  • 27 ⁄8
  • "
  • 18 ⁄4
  • "
  • 55 ⁄4
  • "
  • 19 ⁄2
  • "
  • 3"
  • 32"
  • 48"
  • 8 ⁄4
  • "
  • 107"

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    The Geometry of Nim Arxiv:1109.6712V1 [Math.CO] 30

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    Branes and Polytopes Luca Romano email address: [email protected] ABSTRACT We investigate the hierarchies of half-supersymmetric branes in maximal supergravity theories. By studying the action of the Weyl group of the U-duality group of maximal supergravities we discover a set of universal algebraic rules describing the number of independent 1/2-BPS p-branes, rank by rank, in any dimension. We show that these relations describe the symmetries of certain families of uniform polytopes. This induces a correspondence between half-supersymmetric branes and vertices of opportune uniform polytopes. We show that half-supersymmetric 0-, 1- and 2-branes are in correspondence with the vertices of the k21, 2k1 and 1k2 families of uniform polytopes, respectively, while 3-branes correspond to the vertices of the rectified version of the 2k1 family. For 4-branes and higher rank solutions we find a general behavior. The interpretation of half- supersymmetric solutions as vertices of uniform polytopes reveals some intriguing aspects. One of the most relevant is a triality relation between 0-, 1- and 2-branes. arXiv:1705.01294v1 [hep-th] 3 May 2017 Contents Introduction 2 1 Coxeter Group and Weyl Group 3 1.1 WeylGroup........................................ 6 2 Branes in E11 7 3 Algebraic Structures Behind Half-Supersymmetric Branes 12 4 Branes ad Polytopes 15 Conclusions 27 A Polytopes 30 B Petrie Polygons 30 1 Introduction Since their discovery branes gained a prominent role in the analysis of M-theories and du- alities [1]. One of the most important class of branes consists in Dirichlet branes, or D-branes. D-branes appear in string theory as boundary terms for open strings with mixed Dirichlet-Neumann boundary conditions and, due to their tension, scaling with a negative power of the string cou- pling constant, they are non-perturbative objects [2].
  • Physical Interpretation of the 30 8-Simplexes in the E8 240-Polytope

    Physical Interpretation of the 30 8-Simplexes in the E8 240-Polytope

    Physical Interpretation of the 30 8-simplexes in the E8 240-Polytope: Frank Dodd (Tony) Smith, Jr. 2017 - viXra 1702.0058 248-dim Lie Group E8 has 240 Root Vectors arranged on a 7-sphere S7 in 8-dim space. The 12 vertices of a cuboctahedron live on a 2-sphere S2 in 3-dim space. They are also the 4x3 = 12 outer vertices of 4 tetrahedra (3-simplexes) that share one inner vertex at the center of the cuboctahedron. This paper explores how the 240 vertices of the E8 Polytope in 8-dim space are related to the 30x8 = 240 outer vertices (red in figure below) of 30 8-simplexes whose 9th vertex is a shared inner vertex (yellow in figure below) at the center of the E8 Polytope. The 8-simplex has 9 vertices, 36 edges, 84 triangles, 126 tetrahedron cells, 126 4-simplex faces, 84 5-simplex faces, 36 6-simplex faces, 9 7-simplex faces, and 1 8-dim volume The real 4_21 Witting polytope of the E8 lattice in R8 has 240 vertices; 6,720 edges; 60,480 triangular faces; 241,920 tetrahedra; 483,840 4-simplexes; 483,840 5-simplexes 4_00; 138,240 + 69,120 6-simplexes 4_10 and 4_01; and 17,280 = 2,160x8 7-simplexes 4_20 and 2,160 7-cross-polytopes 4_11. The cuboctahedron corresponds by Jitterbug Transformation to the icosahedron. The 20 2-dim faces of an icosahedon in 3-dim space (image from spacesymmetrystructure.wordpress.com) are also the 20 outer faces of 20 not-exactly-regular-in-3-dim tetrahedra (3-simplexes) that share one inner vertex at the center of the icosahedron, but that correspondence does not extend to the case of 8-simplexes in an E8 polytope, whose faces are both 7-simplexes and 7-cross-polytopes, similar to the cubocahedron, but not its Jitterbug-transform icosahedron with only triangle = 2-simplex faces.
  • 15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm

    15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm

    15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P .
  • Frequently Asked Questions in Polyhedral Computation

    Frequently Asked Questions in Polyhedral Computation

    Frequently Asked Questions in Polyhedral Computation http://www.ifor.math.ethz.ch/~fukuda/polyfaq/polyfaq.html Komei Fukuda Swiss Federal Institute of Technology Lausanne and Zurich, Switzerland [email protected] Version June 18, 2004 Contents 1 What is Polyhedral Computation FAQ? 2 2 Convex Polyhedron 3 2.1 What is convex polytope/polyhedron? . 3 2.2 What are the faces of a convex polytope/polyhedron? . 3 2.3 What is the face lattice of a convex polytope . 4 2.4 What is a dual of a convex polytope? . 4 2.5 What is simplex? . 4 2.6 What is cube/hypercube/cross polytope? . 5 2.7 What is simple/simplicial polytope? . 5 2.8 What is 0-1 polytope? . 5 2.9 What is the best upper bound of the numbers of k-dimensional faces of a d- polytope with n vertices? . 5 2.10 What is convex hull? What is the convex hull problem? . 6 2.11 What is the Minkowski-Weyl theorem for convex polyhedra? . 6 2.12 What is the vertex enumeration problem, and what is the facet enumeration problem? . 7 1 2.13 How can one enumerate all faces of a convex polyhedron? . 7 2.14 What computer models are appropriate for the polyhedral computation? . 8 2.15 How do we measure the complexity of a convex hull algorithm? . 8 2.16 How many facets does the average polytope with n vertices in Rd have? . 9 2.17 How many facets can a 0-1 polytope with n vertices in Rd have? . 10 2.18 How hard is it to verify that an H-polyhedron PH and a V-polyhedron PV are equal? .
  • Sampling Uniformly from the Unit Simplex

    Sampling Uniformly from the Unit Simplex

    Sampling Uniformly from the Unit Simplex Noah A. Smith and Roy W. Tromble Department of Computer Science / Center for Language and Speech Processing Johns Hopkins University {nasmith, royt}@cs.jhu.edu August 2004 Abstract We address the problem of selecting a point from a unit simplex, uniformly at random. This problem is important, for instance, when random multinomial probability distributions are required. We show that a previously proposed algorithm is incorrect, and demonstrate a corrected algorithm. 1 Introduction Suppose we wish to select a multinomial distribution over n events, and we wish to do so uniformly n across the space of such distributions. Such a distribution is characterized by a vector ~p ∈ R such that n X pi = 1 (1) i=1 and pi ≥ 0, ∀i ∈ {1, 2, ..., n} (2) n In practice, of course, we cannot sample from R or even an interval in R; computers have only finite precision. One familiar technique for random generation in real intervals is to select a random integer and normalize it within the desired interval. This easily solves the problem when n = 2; select an integer x uniformly from among {0, 1, 2, ..., M} (where M is, perhaps, the largest integer x x that can be represented), and then let p1 = M and p2 = 1 − M . What does it mean to sample uniformly under this kind of scheme? There are clearly M + 1 discrete distributions from which we sample, each corresponding to a choice of x. If we sample x uniformly from {0, 1, ..., M}, then then we have equal probability of choosing any of these M + 1 distributions.
  • Convex Polytopes and Tilings with Few Flag Orbits

    Convex Polytopes and Tilings with Few Flag Orbits

    Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.